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<div class="body" role="main">
<div class="section" id="module-esys.escript.pdetools">
<span id="esys-escript-pdetools-package"></span><h1>esys.escript.pdetools Package<a class="headerlink" href="#module-esys.escript.pdetools" title="Permalink to this headline">¶</a></h1>
<ul class="simple">
<li></li>
</ul>
<div class="section" id="classes">
<h2>Classes<a class="headerlink" href="#classes" title="Permalink to this headline">¶</a></h2>
<ul class="simple">
<li><a class="reference internal" href="#esys.escript.pdetools.ArithmeticTuple" title="esys.escript.pdetools.ArithmeticTuple"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ArithmeticTuple</span></code></a></li>
<li><a class="reference internal" href="#esys.escript.pdetools.CorrectionFailed" title="esys.escript.pdetools.CorrectionFailed"><code class="xref py py-obj docutils literal notranslate"><span class="pre">CorrectionFailed</span></code></a></li>
<li><a class="reference internal" href="#esys.escript.pdetools.Defect" title="esys.escript.pdetools.Defect"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Defect</span></code></a></li>
<li><a class="reference internal" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem" title="esys.escript.pdetools.HomogeneousSaddlePointProblem"><code class="xref py py-obj docutils literal notranslate"><span class="pre">HomogeneousSaddlePointProblem</span></code></a></li>
<li><a class="reference internal" href="#esys.escript.pdetools.IndefinitePreconditioner" title="esys.escript.pdetools.IndefinitePreconditioner"><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndefinitePreconditioner</span></code></a></li>
<li><a class="reference internal" href="#esys.escript.pdetools.IterationBreakDown" title="esys.escript.pdetools.IterationBreakDown"><code class="xref py py-obj docutils literal notranslate"><span class="pre">IterationBreakDown</span></code></a></li>
<li><a class="reference internal" href="#esys.escript.pdetools.Locator" title="esys.escript.pdetools.Locator"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Locator</span></code></a></li>
<li><a class="reference internal" href="#esys.escript.pdetools.MaxIterReached" title="esys.escript.pdetools.MaxIterReached"><code class="xref py py-obj docutils literal notranslate"><span class="pre">MaxIterReached</span></code></a></li>
<li><a class="reference internal" href="#esys.escript.pdetools.NegativeNorm" title="esys.escript.pdetools.NegativeNorm"><code class="xref py py-obj docutils literal notranslate"><span class="pre">NegativeNorm</span></code></a></li>
<li><a class="reference internal" href="#esys.escript.pdetools.NoPDE" title="esys.escript.pdetools.NoPDE"><code class="xref py py-obj docutils literal notranslate"><span class="pre">NoPDE</span></code></a></li>
<li><a class="reference internal" href="#esys.escript.pdetools.Projector" title="esys.escript.pdetools.Projector"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Projector</span></code></a></li>
<li><a class="reference internal" href="#esys.escript.pdetools.SolverSchemeException" title="esys.escript.pdetools.SolverSchemeException"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SolverSchemeException</span></code></a></li>
<li><a class="reference internal" href="#esys.escript.pdetools.TimeIntegrationManager" title="esys.escript.pdetools.TimeIntegrationManager"><code class="xref py py-obj docutils literal notranslate"><span class="pre">TimeIntegrationManager</span></code></a></li>
</ul>
<dl class="class">
<dt id="esys.escript.pdetools.ArithmeticTuple">
<em class="property">class </em><code class="descclassname">esys.escript.pdetools.</code><code class="descname">ArithmeticTuple</code><span class="sig-paren">(</span><em>*args</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.ArithmeticTuple" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">object</span></code></p>
<p>Tuple supporting inplace update x+=y and scaling x=a*y where <code class="docutils literal notranslate"><span class="pre">x,y</span></code> is an
ArithmeticTuple and <code class="docutils literal notranslate"><span class="pre">a</span></code> is a float.</p>
<p>Example of usage:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">esys.escript</span> <span class="k">import</span> <span class="n">Data</span>
<span class="kn">from</span> <span class="nn">numpy</span> <span class="k">import</span> <span class="n">array</span>
<span class="n">a</span><span class="o">=</span><span class="n">eData</span><span class="p">(</span><span class="o">...</span><span class="p">)</span>
<span class="n">b</span><span class="o">=</span><span class="n">array</span><span class="p">([</span><span class="mf">1.</span><span class="p">,</span><span class="mf">4.</span><span class="p">])</span>
<span class="n">x</span><span class="o">=</span><span class="n">ArithmeticTuple</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">)</span>
<span class="n">y</span><span class="o">=</span><span class="mf">5.</span><span class="o">*</span><span class="n">x</span>
</pre></div>
</div>
<dl class="method">
<dt id="esys.escript.pdetools.ArithmeticTuple.__init__">
<code class="descname">__init__</code><span class="sig-paren">(</span><em>*args</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.ArithmeticTuple.__init__" title="Permalink to this definition">¶</a></dt>
<dd><p>Initializes object with elements <code class="docutils literal notranslate"><span class="pre">args</span></code>.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>args</strong> – tuple of objects that support inplace add (x+=y) and
scaling (x=a*y)</td>
</tr>
</tbody>
</table>
</dd></dl>
</dd></dl>
<dl class="class">
<dt id="esys.escript.pdetools.CorrectionFailed">
<em class="property">class </em><code class="descclassname">esys.escript.pdetools.</code><code class="descname">CorrectionFailed</code><a class="headerlink" href="#esys.escript.pdetools.CorrectionFailed" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">esys.escriptcore.pdetools.SolverSchemeException</span></code></p>
<p>Exception thrown if no convergence has been achieved in the solution
correction scheme.</p>
<dl class="method">
<dt id="esys.escript.pdetools.CorrectionFailed.__init__">
<code class="descname">__init__</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.CorrectionFailed.__init__" title="Permalink to this definition">¶</a></dt>
<dd><p>Initialize self. See help(type(self)) for accurate signature.</p>
</dd></dl>
<dl class="attribute">
<dt id="esys.escript.pdetools.CorrectionFailed.args">
<code class="descname">args</code><a class="headerlink" href="#esys.escript.pdetools.CorrectionFailed.args" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.CorrectionFailed.with_traceback">
<code class="descname">with_traceback</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.CorrectionFailed.with_traceback" title="Permalink to this definition">¶</a></dt>
<dd><p>Exception.with_traceback(tb) –
set self.__traceback__ to tb and return self.</p>
</dd></dl>
</dd></dl>
<dl class="class">
<dt id="esys.escript.pdetools.Defect">
<em class="property">class </em><code class="descclassname">esys.escript.pdetools.</code><code class="descname">Defect</code><a class="headerlink" href="#esys.escript.pdetools.Defect" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">object</span></code></p>
<p>Defines a non-linear defect F(x) of a variable x. This class includes
two functions (bilinearform and eval) that must be overridden by subclassing
before use.</p>
<dl class="method">
<dt id="esys.escript.pdetools.Defect.__init__">
<code class="descname">__init__</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Defect.__init__" title="Permalink to this definition">¶</a></dt>
<dd><p>Initializes defect.</p>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.Defect.bilinearform">
<code class="descname">bilinearform</code><span class="sig-paren">(</span><em>x0</em>, <em>x1</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Defect.bilinearform" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the inner product of x0 and x1</p>
<p>NOTE: MUST BE OVERRIDDEN BY A SUBCLASS</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
<li><strong>x0</strong> – value for x0</li>
<li><strong>x1</strong> – value for x1</li>
</ul>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">the inner product of x0 and x1</p>
</td>
</tr>
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last"><code class="docutils literal notranslate"><span class="pre">float</span></code></p>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.Defect.derivative">
<code class="descname">derivative</code><span class="sig-paren">(</span><em>F0</em>, <em>x0</em>, <em>v</em>, <em>v_is_normalised=True</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Defect.derivative" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the directional derivative at <code class="docutils literal notranslate"><span class="pre">x0</span></code> in the direction of <code class="docutils literal notranslate"><span class="pre">v</span></code>.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
<li><strong>F0</strong> – value of this defect at x0</li>
<li><strong>x0</strong> – value at which derivative is calculated</li>
<li><strong>v</strong> – direction</li>
<li><strong>v_is_normalised</strong> – True to indicate that <code class="docutils literal notranslate"><span class="pre">v</span></code> is nomalized
(self.norm(v)=0)</li>
</ul>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">derivative of this defect at x0 in the direction of <code class="docutils literal notranslate"><span class="pre">v</span></code></p>
</td>
</tr>
<tr class="field-odd field"><th class="field-name">Note:</th><td class="field-body"><p class="first last">by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is
used but this method maybe overwritten to use exact evaluation.</p>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.Defect.eval">
<code class="descname">eval</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Defect.eval" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the value F of a given <code class="docutils literal notranslate"><span class="pre">x</span></code>.</p>
<p>NOTE: MUST BE OVERRIDDEN BY A SUBCLASS</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>x</strong> – value for which the defect <code class="docutils literal notranslate"><span class="pre">F</span></code> is evaluated</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body">value of the defect at <code class="docutils literal notranslate"><span class="pre">x</span></code></td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.Defect.getDerivativeIncrementLength">
<code class="descname">getDerivativeIncrementLength</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Defect.getDerivativeIncrementLength" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the relative increment length used to approximate the
derivative of the defect.
:return: value of the defect at <code class="docutils literal notranslate"><span class="pre">x</span></code>
:rtype: positive <code class="docutils literal notranslate"><span class="pre">float</span></code></p>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.Defect.norm">
<code class="descname">norm</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Defect.norm" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the norm of argument <code class="docutils literal notranslate"><span class="pre">x</span></code>.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>x</strong> – a value</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body">norm of argument x</td>
</tr>
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><code class="docutils literal notranslate"><span class="pre">float</span></code></td>
</tr>
<tr class="field-even field"><th class="field-name">Note:</th><td class="field-body">by default <code class="docutils literal notranslate"><span class="pre">sqrt(self.bilinearform(x,x)</span></code> is returned.</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.Defect.setDerivativeIncrementLength">
<code class="descname">setDerivativeIncrementLength</code><span class="sig-paren">(</span><em>inc=1.4901161193847656e-05</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Defect.setDerivativeIncrementLength" title="Permalink to this definition">¶</a></dt>
<dd><p>Sets the relative length of the increment used to approximate the
derivative of the defect. The increment is inc*norm(x)/norm(v)*v in the
direction of v with x as a starting point.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>inc</strong> (positive <code class="docutils literal notranslate"><span class="pre">float</span></code>) – relative increment length</td>
</tr>
</tbody>
</table>
</dd></dl>
</dd></dl>
<dl class="class">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem">
<em class="property">class </em><code class="descclassname">esys.escript.pdetools.</code><code class="descname">HomogeneousSaddlePointProblem</code><span class="sig-paren">(</span><em>**kwargs</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">object</span></code></p>
<p>This class provides a framework for solving linear homogeneous saddle
point problems of the form:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">*</span><span class="n">Av</span><span class="o">+</span><span class="n">B</span><span class="o">^*</span><span class="n">p</span><span class="o">=</span><span class="n">f</span><span class="o">*</span>
<span class="o">*</span><span class="n">Bv</span> <span class="o">=</span><span class="mi">0</span><span class="o">*</span>
</pre></div>
</div>
<p>for the unknowns <em>v</em> and <em>p</em> and given operators <em>A</em> and <em>B</em> and
given right hand side <em>f</em>. <em>B^*</em> is the adjoint operator of <em>B</em>.
<em>A</em> may depend weakly on <em>v</em> and <em>p</em>.</p>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.__init__">
<code class="descname">__init__</code><span class="sig-paren">(</span><em>**kwargs</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.__init__" title="Permalink to this definition">¶</a></dt>
<dd><p>initializes the saddle point problem</p>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.Bv">
<code class="descname">Bv</code><span class="sig-paren">(</span><em>v</em>, <em>tol</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.Bv" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns Bv with accuracy <code class="xref py py-obj docutils literal notranslate"><span class="pre">tol</span></code> (overwrite)</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body">equal to the type of p</td>
</tr>
<tr class="field-even field"><th class="field-name">Note:</th><td class="field-body">boundary conditions on p should be zero!</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.getAbsoluteTolerance">
<code class="descname">getAbsoluteTolerance</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.getAbsoluteTolerance" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the absolute tolerance.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Returns:</th><td class="field-body">absolute tolerance</td>
</tr>
<tr class="field-even field"><th class="field-name">Return type:</th><td class="field-body"><code class="docutils literal notranslate"><span class="pre">float</span></code></td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.getDV">
<code class="descname">getDV</code><span class="sig-paren">(</span><em>p</em>, <em>v</em>, <em>tol</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.getDV" title="Permalink to this definition">¶</a></dt>
<dd><p>return a correction to the value for a given v and a given p with accuracy <code class="xref py py-obj docutils literal notranslate"><span class="pre">tol</span></code> (overwrite)</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
<li><strong>p</strong> – pressure</li>
<li><strong>v</strong> – pressure</li>
</ul>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">dv given as <em>dv= A^{-1} (f-A v-B^*p)</em></p>
</td>
</tr>
<tr class="field-odd field"><th class="field-name">Note:</th><td class="field-body"><p class="first last">Only <em>A</em> may depend on <em>v</em> and <em>p</em></p>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.getTolerance">
<code class="descname">getTolerance</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.getTolerance" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the relative tolerance.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Returns:</th><td class="field-body">relative tolerance</td>
</tr>
<tr class="field-even field"><th class="field-name">Return type:</th><td class="field-body"><code class="docutils literal notranslate"><span class="pre">float</span></code></td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.inner_p">
<code class="descname">inner_p</code><span class="sig-paren">(</span><em>p0</em>, <em>p1</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.inner_p" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns inner product of p0 and p1 (overwrite).</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
<li><strong>p0</strong> – a pressure</li>
<li><strong>p1</strong> – a pressure</li>
</ul>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">inner product of p0 and p1</p>
</td>
</tr>
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last"><code class="docutils literal notranslate"><span class="pre">float</span></code></p>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.inner_pBv">
<code class="descname">inner_pBv</code><span class="sig-paren">(</span><em>p</em>, <em>Bv</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.inner_pBv" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns inner product of element p and Bv (overwrite).</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
<li><strong>p</strong> – a pressure increment</li>
<li><strong>Bv</strong> – a residual</li>
</ul>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">inner product of element p and Bv</p>
</td>
</tr>
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first"><code class="docutils literal notranslate"><span class="pre">float</span></code></p>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Note:</th><td class="field-body"><p class="first last">used if PCG is applied.</p>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.norm_Bv">
<code class="descname">norm_Bv</code><span class="sig-paren">(</span><em>Bv</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.norm_Bv" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the norm of Bv (overwrite).</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body">equal to the type of p</td>
</tr>
<tr class="field-even field"><th class="field-name">Note:</th><td class="field-body">boundary conditions on p should be zero!</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.norm_p">
<code class="descname">norm_p</code><span class="sig-paren">(</span><em>p</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.norm_p" title="Permalink to this definition">¶</a></dt>
<dd><p>calculates the norm of <code class="docutils literal notranslate"><span class="pre">p</span></code></p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>p</strong> – a pressure</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body">the norm of <code class="docutils literal notranslate"><span class="pre">p</span></code> using the inner product for pressure</td>
</tr>
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><code class="docutils literal notranslate"><span class="pre">float</span></code></td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.norm_v">
<code class="descname">norm_v</code><span class="sig-paren">(</span><em>v</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.norm_v" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the norm of v (overwrite).</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>v</strong> – a velovity</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body">norm of v</td>
</tr>
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body">non-negative <code class="docutils literal notranslate"><span class="pre">float</span></code></td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.resetControlParameters">
<code class="descname">resetControlParameters</code><span class="sig-paren">(</span><em>K_p=1.0</em>, <em>K_v=1.0</em>, <em>rtol_max=0.01</em>, <em>rtol_min=1e-07</em>, <em>chi_max=0.5</em>, <em>reduction_factor=0.3</em>, <em>theta=0.1</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.resetControlParameters" title="Permalink to this definition">¶</a></dt>
<dd><p>sets a control parameter</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
<li><strong>K_p</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>) – initial value for constant to adjust pressure tolerance</li>
<li><strong>K_v</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>) – initial value for constant to adjust velocity tolerance</li>
<li><strong>rtol_max</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>) – maximuim relative tolerance used to calculate presssure and velocity increment.</li>
<li><strong>chi_max</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>) – maximum tolerable converegence rate.</li>
<li><strong>reduction_factor</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>) – reduction factor for adjustment factors.</li>
</ul>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.setAbsoluteTolerance">
<code class="descname">setAbsoluteTolerance</code><span class="sig-paren">(</span><em>tolerance=0.0</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.setAbsoluteTolerance" title="Permalink to this definition">¶</a></dt>
<dd><p>Sets the absolute tolerance.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>tolerance</strong> (non-negative <code class="docutils literal notranslate"><span class="pre">float</span></code>) – tolerance to be used</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.setControlParameter">
<code class="descname">setControlParameter</code><span class="sig-paren">(</span><em>K_p=None</em>, <em>K_v=None</em>, <em>rtol_max=None</em>, <em>rtol_min=None</em>, <em>chi_max=None</em>, <em>reduction_factor=None</em>, <em>theta=None</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.setControlParameter" title="Permalink to this definition">¶</a></dt>
<dd><p>sets a control parameter</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
<li><strong>K_p</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>) – initial value for constant to adjust pressure tolerance</li>
<li><strong>K_v</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>) – initial value for constant to adjust velocity tolerance</li>
<li><strong>rtol_max</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>) – maximuim relative tolerance used to calculate presssure and velocity increment.</li>
<li><strong>chi_max</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>) – maximum tolerable converegence rate.</li>
</ul>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.setTolerance">
<code class="descname">setTolerance</code><span class="sig-paren">(</span><em>tolerance=0.0001</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.setTolerance" title="Permalink to this definition">¶</a></dt>
<dd><p>Sets the relative tolerance for (v,p).</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>tolerance</strong> (non-negative <code class="docutils literal notranslate"><span class="pre">float</span></code>) – tolerance to be used</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.solve">
<code class="descname">solve</code><span class="sig-paren">(</span><em>v</em>, <em>p</em>, <em>max_iter=20</em>, <em>verbose=False</em>, <em>usePCG=True</em>, <em>iter_restart=20</em>, <em>max_correction_steps=10</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.solve" title="Permalink to this definition">¶</a></dt>
<dd><p>Solves the saddle point problem using initial guesses v and p.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
<li><strong>v</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Data</span></code>) – initial guess for velocity</li>
<li><strong>p</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Data</span></code>) – initial guess for pressure</li>
<li><strong>usePCG</strong> (<code class="docutils literal notranslate"><span class="pre">bool</span></code>) – indicates the usage of the PCG rather than GMRES scheme.</li>
<li><strong>max_iter</strong> (<code class="docutils literal notranslate"><span class="pre">int</span></code>) – maximum number of iteration steps per correction
attempt</li>
<li><strong>verbose</strong> (<code class="docutils literal notranslate"><span class="pre">bool</span></code>) – if True, shows information on the progress of the
saddlepoint problem solver.</li>
<li><strong>iter_restart</strong> (<code class="docutils literal notranslate"><span class="pre">int</span></code>) – restart the iteration after <code class="docutils literal notranslate"><span class="pre">iter_restart</span></code> steps
(only used if useUzaw=False)</li>
</ul>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Return type:</th><td class="field-body"><p class="first"><code class="docutils literal notranslate"><span class="pre">tuple</span></code> of <code class="xref py py-obj docutils literal notranslate"><span class="pre">Data</span></code> objects</p>
</td>
</tr>
<tr class="field-odd field"><th class="field-name">Note:</th><td class="field-body"><p class="first last">typically this method is overwritten by a subclass. It provides a wrapper for the <code class="docutils literal notranslate"><span class="pre">_solve</span></code> method.</p>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.solve_AinvBt">
<code class="descname">solve_AinvBt</code><span class="sig-paren">(</span><em>dp</em>, <em>tol</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.solve_AinvBt" title="Permalink to this definition">¶</a></dt>
<dd><p>Solves <em>A dv=B^*dp</em> with accuracy <code class="xref py py-obj docutils literal notranslate"><span class="pre">tol</span></code></p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>dp</strong> – a pressure increment</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body">the solution of <em>A dv=B^*dp</em></td>
</tr>
<tr class="field-odd field"><th class="field-name">Note:</th><td class="field-body">boundary conditions on dv should be zero! <em>A</em> is the operator used in <code class="docutils literal notranslate"><span class="pre">getDV</span></code> and must not be altered.</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.HomogeneousSaddlePointProblem.solve_prec">
<code class="descname">solve_prec</code><span class="sig-paren">(</span><em>Bv</em>, <em>tol</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.HomogeneousSaddlePointProblem.solve_prec" title="Permalink to this definition">¶</a></dt>
<dd><p>Provides a preconditioner for <em>(BA^{-1}B^ * )</em> applied to Bv with accuracy <code class="xref py py-obj docutils literal notranslate"><span class="pre">tol</span></code></p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body">equal to the type of p</td>
</tr>
<tr class="field-even field"><th class="field-name">Note:</th><td class="field-body">boundary conditions on p should be zero!</td>
</tr>
</tbody>
</table>
</dd></dl>
</dd></dl>
<dl class="class">
<dt id="esys.escript.pdetools.IndefinitePreconditioner">
<em class="property">class </em><code class="descclassname">esys.escript.pdetools.</code><code class="descname">IndefinitePreconditioner</code><a class="headerlink" href="#esys.escript.pdetools.IndefinitePreconditioner" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">esys.escriptcore.pdetools.SolverSchemeException</span></code></p>
<p>Exception thrown if the preconditioner is not positive definite.</p>
<dl class="method">
<dt id="esys.escript.pdetools.IndefinitePreconditioner.__init__">
<code class="descname">__init__</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.IndefinitePreconditioner.__init__" title="Permalink to this definition">¶</a></dt>
<dd><p>Initialize self. See help(type(self)) for accurate signature.</p>
</dd></dl>
<dl class="attribute">
<dt id="esys.escript.pdetools.IndefinitePreconditioner.args">
<code class="descname">args</code><a class="headerlink" href="#esys.escript.pdetools.IndefinitePreconditioner.args" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.IndefinitePreconditioner.with_traceback">
<code class="descname">with_traceback</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.IndefinitePreconditioner.with_traceback" title="Permalink to this definition">¶</a></dt>
<dd><p>Exception.with_traceback(tb) –
set self.__traceback__ to tb and return self.</p>
</dd></dl>
</dd></dl>
<dl class="class">
<dt id="esys.escript.pdetools.IterationBreakDown">
<em class="property">class </em><code class="descclassname">esys.escript.pdetools.</code><code class="descname">IterationBreakDown</code><a class="headerlink" href="#esys.escript.pdetools.IterationBreakDown" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">esys.escriptcore.pdetools.SolverSchemeException</span></code></p>
<p>Exception thrown if the iteration scheme encountered an incurable breakdown.</p>
<dl class="method">
<dt id="esys.escript.pdetools.IterationBreakDown.__init__">
<code class="descname">__init__</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.IterationBreakDown.__init__" title="Permalink to this definition">¶</a></dt>
<dd><p>Initialize self. See help(type(self)) for accurate signature.</p>
</dd></dl>
<dl class="attribute">
<dt id="esys.escript.pdetools.IterationBreakDown.args">
<code class="descname">args</code><a class="headerlink" href="#esys.escript.pdetools.IterationBreakDown.args" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.IterationBreakDown.with_traceback">
<code class="descname">with_traceback</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.IterationBreakDown.with_traceback" title="Permalink to this definition">¶</a></dt>
<dd><p>Exception.with_traceback(tb) –
set self.__traceback__ to tb and return self.</p>
</dd></dl>
</dd></dl>
<dl class="class">
<dt id="esys.escript.pdetools.Locator">
<em class="property">class </em><code class="descclassname">esys.escript.pdetools.</code><code class="descname">Locator</code><span class="sig-paren">(</span><em>where</em>, <em>x=array([0.</em>, <em>0.</em>, <em>0.])</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Locator" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">object</span></code></p>
<p>Locator provides access to the values of data objects at a given spatial
coordinate x.</p>
<p>In fact, a Locator object finds the sample in the set of samples of a
given function space or domain which is closest to the given point x.</p>
<dl class="method">
<dt id="esys.escript.pdetools.Locator.__init__">
<code class="descname">__init__</code><span class="sig-paren">(</span><em>where</em>, <em>x=array([0.</em>, <em>0.</em>, <em>0.])</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Locator.__init__" title="Permalink to this definition">¶</a></dt>
<dd><p>Initializes a Locator to access values in Data objects on the Doamin
or FunctionSpace for the sample point which is closest to the given
point x.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
<li><strong>where</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">escript.FunctionSpace</span></code>) – function space</li>
<li><strong>x</strong> (<code class="docutils literal notranslate"><span class="pre">numpy.ndarray</span></code> or <code class="docutils literal notranslate"><span class="pre">list</span></code> of <code class="docutils literal notranslate"><span class="pre">numpy.ndarray</span></code>) – location(s) of the Locator</li>
</ul>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.Locator.getFunctionSpace">
<code class="descname">getFunctionSpace</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Locator.getFunctionSpace" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the function space of the Locator.</p>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.Locator.getId">
<code class="descname">getId</code><span class="sig-paren">(</span><em>item=None</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Locator.getId" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the identifier of the location.</p>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.Locator.getValue">
<code class="descname">getValue</code><span class="sig-paren">(</span><em>data</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Locator.getValue" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the value of <code class="docutils literal notranslate"><span class="pre">data</span></code> at the Locator if <code class="docutils literal notranslate"><span class="pre">data</span></code> is a <code class="xref py py-obj docutils literal notranslate"><span class="pre">Data</span></code>
object otherwise the object is returned.</p>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.Locator.getX">
<code class="descname">getX</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Locator.getX" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the exact coordinates of the Locator.</p>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.Locator.setValue">
<code class="descname">setValue</code><span class="sig-paren">(</span><em>data</em>, <em>v</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Locator.setValue" title="Permalink to this definition">¶</a></dt>
<dd><p>Sets the value of the <code class="docutils literal notranslate"><span class="pre">data</span></code> at the Locator.</p>
</dd></dl>
</dd></dl>
<dl class="class">
<dt id="esys.escript.pdetools.MaxIterReached">
<em class="property">class </em><code class="descclassname">esys.escript.pdetools.</code><code class="descname">MaxIterReached</code><a class="headerlink" href="#esys.escript.pdetools.MaxIterReached" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">esys.escriptcore.pdetools.SolverSchemeException</span></code></p>
<p>Exception thrown if the maximum number of iteration steps is reached.</p>
<dl class="method">
<dt id="esys.escript.pdetools.MaxIterReached.__init__">
<code class="descname">__init__</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.MaxIterReached.__init__" title="Permalink to this definition">¶</a></dt>
<dd><p>Initialize self. See help(type(self)) for accurate signature.</p>
</dd></dl>
<dl class="attribute">
<dt id="esys.escript.pdetools.MaxIterReached.args">
<code class="descname">args</code><a class="headerlink" href="#esys.escript.pdetools.MaxIterReached.args" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.MaxIterReached.with_traceback">
<code class="descname">with_traceback</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.MaxIterReached.with_traceback" title="Permalink to this definition">¶</a></dt>
<dd><p>Exception.with_traceback(tb) –
set self.__traceback__ to tb and return self.</p>
</dd></dl>
</dd></dl>
<dl class="class">
<dt id="esys.escript.pdetools.NegativeNorm">
<em class="property">class </em><code class="descclassname">esys.escript.pdetools.</code><code class="descname">NegativeNorm</code><a class="headerlink" href="#esys.escript.pdetools.NegativeNorm" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">esys.escriptcore.pdetools.SolverSchemeException</span></code></p>
<p>Exception thrown if a norm calculation returns a negative norm.</p>
<dl class="method">
<dt id="esys.escript.pdetools.NegativeNorm.__init__">
<code class="descname">__init__</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.NegativeNorm.__init__" title="Permalink to this definition">¶</a></dt>
<dd><p>Initialize self. See help(type(self)) for accurate signature.</p>
</dd></dl>
<dl class="attribute">
<dt id="esys.escript.pdetools.NegativeNorm.args">
<code class="descname">args</code><a class="headerlink" href="#esys.escript.pdetools.NegativeNorm.args" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.NegativeNorm.with_traceback">
<code class="descname">with_traceback</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.NegativeNorm.with_traceback" title="Permalink to this definition">¶</a></dt>
<dd><p>Exception.with_traceback(tb) –
set self.__traceback__ to tb and return self.</p>
</dd></dl>
</dd></dl>
<dl class="class">
<dt id="esys.escript.pdetools.NoPDE">
<em class="property">class </em><code class="descclassname">esys.escript.pdetools.</code><code class="descname">NoPDE</code><span class="sig-paren">(</span><em>domain</em>, <em>D=None</em>, <em>Y=None</em>, <em>q=None</em>, <em>r=None</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.NoPDE" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">object</span></code></p>
<p>Solves the following problem for u:</p>
<p><em>kronecker[i,j]*D[j]*u[j]=Y[i]</em></p>
<p>with constraint</p>
<p><em>u[j]=r[j]</em> where <em>q[j]>0</em></p>
<p>where <em>D</em>, <em>Y</em>, <em>r</em> and <em>q</em> are given functions of rank 1.</p>
<p>In the case of scalars this takes the form</p>
<p><em>D*u=Y</em></p>
<p>with constraint</p>
<p><em>u=r</em> where <em>q>0</em></p>
<p>where <em>D</em>, <em>Y</em>, <em>r</em> and <em>q</em> are given scalar functions.</p>
<p>The constraint overwrites any other condition.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Note:</th><td class="field-body">This class is similar to the <code class="xref py py-obj docutils literal notranslate"><span class="pre">linearPDEs.LinearPDE</span></code> class with
A=B=C=X=0 but has the intention that all input parameters are given
in <code class="xref py py-obj docutils literal notranslate"><span class="pre">Solution</span></code> or <code class="xref py py-obj docutils literal notranslate"><span class="pre">ReducedSolution</span></code>.</td>
</tr>
</tbody>
</table>
<dl class="method">
<dt id="esys.escript.pdetools.NoPDE.__init__">
<code class="descname">__init__</code><span class="sig-paren">(</span><em>domain</em>, <em>D=None</em>, <em>Y=None</em>, <em>q=None</em>, <em>r=None</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.NoPDE.__init__" title="Permalink to this definition">¶</a></dt>
<dd><p>Initializes the problem.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
<li><strong>domain</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">Domain</span></code>) – domain of the PDE</li>
<li><strong>D</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>, <code class="docutils literal notranslate"><span class="pre">int</span></code>, <code class="docutils literal notranslate"><span class="pre">numpy.ndarray</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">Data</span></code>) – coefficient of the solution</li>
<li><strong>Y</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>, <code class="docutils literal notranslate"><span class="pre">int</span></code>, <code class="docutils literal notranslate"><span class="pre">numpy.ndarray</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">Data</span></code>) – right hand side</li>
<li><strong>q</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>, <code class="docutils literal notranslate"><span class="pre">int</span></code>, <code class="docutils literal notranslate"><span class="pre">numpy.ndarray</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">Data</span></code>) – location of constraints</li>
<li><strong>r</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>, <code class="docutils literal notranslate"><span class="pre">int</span></code>, <code class="docutils literal notranslate"><span class="pre">numpy.ndarray</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">Data</span></code>) – value of solution at locations of constraints</li>
</ul>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.NoPDE.getSolution">
<code class="descname">getSolution</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.NoPDE.getSolution" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the solution.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Returns:</th><td class="field-body">the solution of the problem</td>
</tr>
<tr class="field-even field"><th class="field-name">Return type:</th><td class="field-body"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Data</span></code> object in the <code class="xref py py-obj docutils literal notranslate"><span class="pre">FunctionSpace</span></code> <code class="xref py py-obj docutils literal notranslate"><span class="pre">Solution</span></code> or
<code class="xref py py-obj docutils literal notranslate"><span class="pre">ReducedSolution</span></code></td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.NoPDE.setReducedOff">
<code class="descname">setReducedOff</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.NoPDE.setReducedOff" title="Permalink to this definition">¶</a></dt>
<dd><p>Sets the <code class="xref py py-obj docutils literal notranslate"><span class="pre">FunctionSpace</span></code> of the solution to <code class="xref py py-obj docutils literal notranslate"><span class="pre">Solution</span></code>.</p>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.NoPDE.setReducedOn">
<code class="descname">setReducedOn</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.NoPDE.setReducedOn" title="Permalink to this definition">¶</a></dt>
<dd><p>Sets the <code class="xref py py-obj docutils literal notranslate"><span class="pre">FunctionSpace</span></code> of the solution to <code class="xref py py-obj docutils literal notranslate"><span class="pre">ReducedSolution</span></code>.</p>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.NoPDE.setValue">
<code class="descname">setValue</code><span class="sig-paren">(</span><em>D=None</em>, <em>Y=None</em>, <em>q=None</em>, <em>r=None</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.NoPDE.setValue" title="Permalink to this definition">¶</a></dt>
<dd><p>Assigns values to the parameters.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
<li><strong>D</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>, <code class="docutils literal notranslate"><span class="pre">int</span></code>, <code class="docutils literal notranslate"><span class="pre">numpy.ndarray</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">Data</span></code>) – coefficient of the solution</li>
<li><strong>Y</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>, <code class="docutils literal notranslate"><span class="pre">int</span></code>, <code class="docutils literal notranslate"><span class="pre">numpy.ndarray</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">Data</span></code>) – right hand side</li>
<li><strong>q</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>, <code class="docutils literal notranslate"><span class="pre">int</span></code>, <code class="docutils literal notranslate"><span class="pre">numpy.ndarray</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">Data</span></code>) – location of constraints</li>
<li><strong>r</strong> (<code class="docutils literal notranslate"><span class="pre">float</span></code>, <code class="docutils literal notranslate"><span class="pre">int</span></code>, <code class="docutils literal notranslate"><span class="pre">numpy.ndarray</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">Data</span></code>) – value of solution at locations of constraints</li>
</ul>
</td>
</tr>
</tbody>
</table>
</dd></dl>
</dd></dl>
<dl class="class">
<dt id="esys.escript.pdetools.Projector">
<em class="property">class </em><code class="descclassname">esys.escript.pdetools.</code><code class="descname">Projector</code><span class="sig-paren">(</span><em>domain</em>, <em>reduce=True</em>, <em>fast=True</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Projector" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">object</span></code></p>
<p>The Projector is a factory which projects a discontinuous function onto a
continuous function on a given domain.</p>
<dl class="method">
<dt id="esys.escript.pdetools.Projector.__init__">
<code class="descname">__init__</code><span class="sig-paren">(</span><em>domain</em>, <em>reduce=True</em>, <em>fast=True</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Projector.__init__" title="Permalink to this definition">¶</a></dt>
<dd><p>Creates a continuous function space projector for a domain.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
<li><strong>domain</strong> – Domain of the projection.</li>
<li><strong>reduce</strong> – Flag to reduce projection order</li>
<li><strong>fast</strong> – Flag to use a fast method based on matrix lumping</li>
</ul>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.Projector.getSolverOptions">
<code class="descname">getSolverOptions</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Projector.getSolverOptions" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the solver options of the PDE solver.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linearPDEs.SolverOptions</span></code></td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.Projector.getValue">
<code class="descname">getValue</code><span class="sig-paren">(</span><em>input_data</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.Projector.getValue" title="Permalink to this definition">¶</a></dt>
<dd><p>Projects <code class="docutils literal notranslate"><span class="pre">input_data</span></code> onto a continuous function.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>input_data</strong> – the data to be projected</td>
</tr>
</tbody>
</table>
</dd></dl>
</dd></dl>
<dl class="class">
<dt id="esys.escript.pdetools.SolverSchemeException">
<em class="property">class </em><code class="descclassname">esys.escript.pdetools.</code><code class="descname">SolverSchemeException</code><a class="headerlink" href="#esys.escript.pdetools.SolverSchemeException" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">Exception</span></code></p>
<p>This is a generic exception thrown by solvers.</p>
<dl class="method">
<dt id="esys.escript.pdetools.SolverSchemeException.__init__">
<code class="descname">__init__</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.SolverSchemeException.__init__" title="Permalink to this definition">¶</a></dt>
<dd><p>Initialize self. See help(type(self)) for accurate signature.</p>
</dd></dl>
<dl class="attribute">
<dt id="esys.escript.pdetools.SolverSchemeException.args">
<code class="descname">args</code><a class="headerlink" href="#esys.escript.pdetools.SolverSchemeException.args" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.SolverSchemeException.with_traceback">
<code class="descname">with_traceback</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.SolverSchemeException.with_traceback" title="Permalink to this definition">¶</a></dt>
<dd><p>Exception.with_traceback(tb) –
set self.__traceback__ to tb and return self.</p>
</dd></dl>
</dd></dl>
<dl class="class">
<dt id="esys.escript.pdetools.TimeIntegrationManager">
<em class="property">class </em><code class="descclassname">esys.escript.pdetools.</code><code class="descname">TimeIntegrationManager</code><span class="sig-paren">(</span><em>*inital_values</em>, <em>**kwargs</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.TimeIntegrationManager" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">object</span></code></p>
<p>A simple mechanism to manage time dependend values.</p>
<p>Typical usage is:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">dt</span><span class="o">=</span><span class="mf">0.1</span> <span class="c1"># time increment</span>
<span class="n">tm</span><span class="o">=</span><span class="n">TimeIntegrationManager</span><span class="p">(</span><span class="n">inital_value</span><span class="p">,</span><span class="n">p</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="k">while</span> <span class="n">t</span><span class="o"><</span><span class="mf">1.</span>
<span class="n">v_guess</span><span class="o">=</span><span class="n">tm</span><span class="o">.</span><span class="n">extrapolate</span><span class="p">(</span><span class="n">dt</span><span class="p">)</span> <span class="c1"># extrapolate to t+dt</span>
<span class="n">v</span><span class="o">=...</span>
<span class="n">tm</span><span class="o">.</span><span class="n">checkin</span><span class="p">(</span><span class="n">dt</span><span class="p">,</span><span class="n">v</span><span class="p">)</span>
<span class="n">t</span><span class="o">+=</span><span class="n">dt</span>
</pre></div>
</div>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Note:</th><td class="field-body">currently only p=1 is supported.</td>
</tr>
</tbody>
</table>
<dl class="method">
<dt id="esys.escript.pdetools.TimeIntegrationManager.__init__">
<code class="descname">__init__</code><span class="sig-paren">(</span><em>*inital_values</em>, <em>**kwargs</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.TimeIntegrationManager.__init__" title="Permalink to this definition">¶</a></dt>
<dd><p>Sets up the value manager where <code class="docutils literal notranslate"><span class="pre">inital_values</span></code> are the initial values
and p is the order used for extrapolation.</p>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.TimeIntegrationManager.checkin">
<code class="descname">checkin</code><span class="sig-paren">(</span><em>dt</em>, <em>*values</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.TimeIntegrationManager.checkin" title="Permalink to this definition">¶</a></dt>
<dd><p>Adds new values to the manager. The p+1 last values are lost.</p>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.TimeIntegrationManager.extrapolate">
<code class="descname">extrapolate</code><span class="sig-paren">(</span><em>dt</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.TimeIntegrationManager.extrapolate" title="Permalink to this definition">¶</a></dt>
<dd><p>Extrapolates to <code class="docutils literal notranslate"><span class="pre">dt</span></code> forward in time.</p>
</dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.TimeIntegrationManager.getTime">
<code class="descname">getTime</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.TimeIntegrationManager.getTime" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="esys.escript.pdetools.TimeIntegrationManager.getValue">
<code class="descname">getValue</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.TimeIntegrationManager.getValue" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
</dd></dl>
</div>
<div class="section" id="functions">
<h2>Functions<a class="headerlink" href="#functions" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="esys.escript.pdetools.BoundaryValuesFromVolumeTag">
<code class="descclassname">esys.escript.pdetools.</code><code class="descname">BoundaryValuesFromVolumeTag</code><span class="sig-paren">(</span><em>domain</em>, <em>**values</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.BoundaryValuesFromVolumeTag" title="Permalink to this definition">¶</a></dt>
<dd><p>Creates a mask on the Solution(domain) function space where the value is
one for samples that touch regions tagged by tags.</p>
<p>Usage: m=BoundaryValuesFromVolumeTag(domain, ham=1, f=6)</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>domain</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">escript.Domain</span></code>) – domain to be used</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body">a mask which marks samples that are touching the boundary tagged
by any of the given tags</td>
</tr>
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><code class="xref py py-obj docutils literal notranslate"><span class="pre">escript.Data</span></code> of rank 0</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="function">
<dt id="esys.escript.pdetools.GMRES">
<code class="descclassname">esys.escript.pdetools.</code><code class="descname">GMRES</code><span class="sig-paren">(</span><em>r</em>, <em>Aprod</em>, <em>x</em>, <em>bilinearform</em>, <em>atol=0</em>, <em>rtol=1e-08</em>, <em>iter_max=100</em>, <em>iter_restart=20</em>, <em>verbose=False</em>, <em>P_R=None</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.GMRES" title="Permalink to this definition">¶</a></dt>
<dd><p>Solver for</p>
<p><em>Ax=b</em></p>
<p>with a general operator A (more details required!).
It uses the generalized minimum residual method (GMRES).</p>
<p>The iteration is terminated if</p>
<p><em>|r| <= atol+rtol*|r0|</em></p>
<p>where <em>r0</em> is the initial residual and <em>|.|</em> is the energy norm. In fact</p>
<p><em>|r| = sqrt( bilinearform(r,r))</em></p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
<li><strong>r</strong> (<em>any object supporting inplace add</em><em> (</em><em>x+=y</em><em>) </em><em>and scaling</em><em> (</em><em>x=scalar*y</em><em>)</em>) – initial residual <em>r=b-Ax</em>. <code class="docutils literal notranslate"><span class="pre">r</span></code> is altered.</li>
<li><strong>x</strong> (same like <code class="docutils literal notranslate"><span class="pre">r</span></code>) – an initial guess for the solution</li>
<li><strong>Aprod</strong> (function <code class="docutils literal notranslate"><span class="pre">Aprod(x)</span></code> where <code class="docutils literal notranslate"><span class="pre">x</span></code> is of the same object like
argument <code class="docutils literal notranslate"><span class="pre">x</span></code>. The returned object needs to be of the same
type like argument <code class="docutils literal notranslate"><span class="pre">r</span></code>.) – returns the value Ax</li>
<li><strong>bilinearform</strong> (function <code class="docutils literal notranslate"><span class="pre">bilinearform(x,r)</span></code> where <code class="docutils literal notranslate"><span class="pre">x</span></code> is of the same
type like argument <code class="docutils literal notranslate"><span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">r</span></code>. The returned value is
a <code class="docutils literal notranslate"><span class="pre">float</span></code>.) – inner product <code class="docutils literal notranslate"><span class="pre"><x,r></span></code></li>
<li><strong>atol</strong> (non-negative <code class="docutils literal notranslate"><span class="pre">float</span></code>) – absolute tolerance</li>
<li><strong>rtol</strong> (non-negative <code class="docutils literal notranslate"><span class="pre">float</span></code>) – relative tolerance</li>
<li><strong>iter_max</strong> (<code class="docutils literal notranslate"><span class="pre">int</span></code>) – maximum number of iteration steps</li>
<li><strong>iter_restart</strong> (<code class="docutils literal notranslate"><span class="pre">int</span></code>) – in order to save memory the orthogonalization process
is terminated after <code class="docutils literal notranslate"><span class="pre">iter_restart</span></code> steps and the
iteration is restarted.</li>
</ul>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">the solution approximation and the corresponding residual</p>
</td>
</tr>
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first"><code class="docutils literal notranslate"><span class="pre">tuple</span></code></p>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Warning:</th><td class="field-body"><p class="first last"><code class="docutils literal notranslate"><span class="pre">r</span></code> and <code class="docutils literal notranslate"><span class="pre">x</span></code> are altered.</p>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="function">
<dt id="esys.escript.pdetools.MINRES">
<code class="descclassname">esys.escript.pdetools.</code><code class="descname">MINRES</code><span class="sig-paren">(</span><em>r</em>, <em>Aprod</em>, <em>x</em>, <em>Msolve</em>, <em>bilinearform</em>, <em>atol=0</em>, <em>rtol=1e-08</em>, <em>iter_max=100</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.MINRES" title="Permalink to this definition">¶</a></dt>
<dd><p>Solver for</p>
<p><em>Ax=b</em></p>
<p>with a symmetric and positive definite operator A (more details required!).
It uses the minimum residual method (MINRES) with preconditioner M
providing an approximation of A.</p>
<p>The iteration is terminated if</p>
<p><em>|r| <= atol+rtol*|r0|</em></p>
<p>where <em>r0</em> is the initial residual and <em>|.|</em> is the energy norm. In fact</p>
<p><em>|r| = sqrt( bilinearform(Msolve(r),r))</em></p>
<p>For details on the preconditioned conjugate gradient method see the book:</p>
<p>“Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
C. Romine, and H. van der Vorst”.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
<li><strong>r</strong> (<em>any object supporting inplace add</em><em> (</em><em>x+=y</em><em>) </em><em>and scaling</em><em> (</em><em>x=scalar*y</em><em>)</em>) – initial residual <em>r=b-Ax</em>. <code class="docutils literal notranslate"><span class="pre">r</span></code> is altered.</li>
<li><strong>x</strong> (<em>any object supporting inplace add</em><em> (</em><em>x+=y</em><em>) </em><em>and scaling</em><em> (</em><em>x=scalar*y</em><em>)</em>) – an initial guess for the solution</li>
<li><strong>Aprod</strong> (function <code class="docutils literal notranslate"><span class="pre">Aprod(x)</span></code> where <code class="docutils literal notranslate"><span class="pre">x</span></code> is of the same object like
argument <code class="docutils literal notranslate"><span class="pre">x</span></code>. The returned object needs to be of the same
type like argument <code class="docutils literal notranslate"><span class="pre">r</span></code>.) – returns the value Ax</li>
<li><strong>Msolve</strong> (function <code class="docutils literal notranslate"><span class="pre">Msolve(r)</span></code> where <code class="docutils literal notranslate"><span class="pre">r</span></code> is of the same type like
argument <code class="docutils literal notranslate"><span class="pre">r</span></code>. The returned object needs to be of the same
type like argument <code class="docutils literal notranslate"><span class="pre">x</span></code>.) – solves Mx=r</li>
<li><strong>bilinearform</strong> (function <code class="docutils literal notranslate"><span class="pre">bilinearform(x,r)</span></code> where <code class="docutils literal notranslate"><span class="pre">x</span></code> is of the same
type like argument <code class="docutils literal notranslate"><span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">r</span></code> is. The returned value
is a <code class="docutils literal notranslate"><span class="pre">float</span></code>.) – inner product <code class="docutils literal notranslate"><span class="pre"><x,r></span></code></li>
<li><strong>atol</strong> (non-negative <code class="docutils literal notranslate"><span class="pre">float</span></code>) – absolute tolerance</li>
<li><strong>rtol</strong> (non-negative <code class="docutils literal notranslate"><span class="pre">float</span></code>) – relative tolerance</li>
<li><strong>iter_max</strong> (<code class="docutils literal notranslate"><span class="pre">int</span></code>) – maximum number of iteration steps</li>
</ul>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">the solution approximation and the corresponding residual</p>
</td>
</tr>
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first"><code class="docutils literal notranslate"><span class="pre">tuple</span></code></p>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Warning:</th><td class="field-body"><p class="first last"><code class="docutils literal notranslate"><span class="pre">r</span></code> and <code class="docutils literal notranslate"><span class="pre">x</span></code> are altered.</p>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="function">
<dt id="esys.escript.pdetools.MaskFromBoundaryTag">
<code class="descclassname">esys.escript.pdetools.</code><code class="descname">MaskFromBoundaryTag</code><span class="sig-paren">(</span><em>domain</em>, <em>*tags</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.MaskFromBoundaryTag" title="Permalink to this definition">¶</a></dt>
<dd><p>Creates a mask on the Solution(domain) function space where the value is
one for samples that touch the boundary tagged by tags.</p>
<p>Usage: m=MaskFromBoundaryTag(domain, “left”, “right”)</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
<li><strong>domain</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">escript.Domain</span></code>) – domain to be used</li>
<li><strong>tags</strong> (<code class="docutils literal notranslate"><span class="pre">str</span></code>) – boundary tags</li>
</ul>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">a mask which marks samples that are touching the boundary tagged
by any of the given tags</p>
</td>
</tr>
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last"><code class="xref py py-obj docutils literal notranslate"><span class="pre">escript.Data</span></code> of rank 0</p>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="function">
<dt id="esys.escript.pdetools.MaskFromTag">
<code class="descclassname">esys.escript.pdetools.</code><code class="descname">MaskFromTag</code><span class="sig-paren">(</span><em>domain</em>, <em>*tags</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.MaskFromTag" title="Permalink to this definition">¶</a></dt>
<dd><p>Creates a mask on the Solution(domain) function space where the value is
one for samples that touch regions tagged by tags.</p>
<p>Usage: m=MaskFromTag(domain, “ham”)</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
<li><strong>domain</strong> (<code class="xref py py-obj docutils literal notranslate"><span class="pre">escript.Domain</span></code>) – domain to be used</li>
<li><strong>tags</strong> (<code class="docutils literal notranslate"><span class="pre">str</span></code>) – boundary tags</li>
</ul>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">a mask which marks samples that are touching the boundary tagged
by any of the given tags</p>
</td>
</tr>
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last"><code class="xref py py-obj docutils literal notranslate"><span class="pre">escript.Data</span></code> of rank 0</p>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="function">
<dt id="esys.escript.pdetools.NewtonGMRES">
<code class="descclassname">esys.escript.pdetools.</code><code class="descname">NewtonGMRES</code><span class="sig-paren">(</span><em>defect</em>, <em>x</em>, <em>iter_max=100</em>, <em>sub_iter_max=20</em>, <em>atol=0</em>, <em>rtol=0.0001</em>, <em>subtol_max=0.5</em>, <em>gamma=0.9</em>, <em>verbose=False</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.NewtonGMRES" title="Permalink to this definition">¶</a></dt>
<dd><p>Solves a non-linear problem <em>F(x)=0</em> for unknown <em>x</em> using the stopping
criterion:</p>
<p><em>norm(F(x) <= atol + rtol * norm(F(x0)</em></p>
<p>where <em>x0</em> is the initial guess.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
<li><strong>defect</strong> (<a class="reference internal" href="#esys.escript.pdetools.Defect" title="esys.escript.pdetools.Defect"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Defect</span></code></a>) – object defining the function <em>F</em>. <code class="docutils literal notranslate"><span class="pre">defect.norm</span></code> defines the
<em>norm</em> used in the stopping criterion.</li>
<li><strong>x</strong> (any object type allowing basic operations such as
<code class="docutils literal notranslate"><span class="pre">numpy.ndarray</span></code>, <code class="xref py py-obj docutils literal notranslate"><span class="pre">Data</span></code>) – initial guess for the solution, <code class="docutils literal notranslate"><span class="pre">x</span></code> is altered.</li>
<li><strong>iter_max</strong> (positive <code class="docutils literal notranslate"><span class="pre">int</span></code>) – maximum number of iteration steps</li>
<li><strong>sub_iter_max</strong> (positive <code class="docutils literal notranslate"><span class="pre">int</span></code>) – maximum number of inner iteration steps</li>
<li><strong>atol</strong> (positive <code class="docutils literal notranslate"><span class="pre">float</span></code>) – absolute tolerance for the solution</li>
<li><strong>rtol</strong> (positive <code class="docutils literal notranslate"><span class="pre">float</span></code>) – relative tolerance for the solution</li>
<li><strong>gamma</strong> (positive <code class="docutils literal notranslate"><span class="pre">float</span></code>, less than 1) – tolerance safety factor for inner iteration</li>
<li><strong>subtol_max</strong> (positive <code class="docutils literal notranslate"><span class="pre">float</span></code>, less than 1) – upper bound for inner tolerance</li>
</ul>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">an approximation of the solution with the desired accuracy</p>
</td>
</tr>
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">same type as the initial guess</p>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="function">
<dt id="esys.escript.pdetools.PCG">
<code class="descclassname">esys.escript.pdetools.</code><code class="descname">PCG</code><span class="sig-paren">(</span><em>r</em>, <em>Aprod</em>, <em>x</em>, <em>Msolve</em>, <em>bilinearform</em>, <em>atol=0</em>, <em>rtol=1e-08</em>, <em>iter_max=100</em>, <em>initial_guess=True</em>, <em>verbose=False</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.PCG" title="Permalink to this definition">¶</a></dt>
<dd><p>Solver for</p>
<p><em>Ax=b</em></p>
<p>with a symmetric and positive definite operator A (more details required!).
It uses the conjugate gradient method with preconditioner M providing an
approximation of A.</p>
<p>The iteration is terminated if</p>
<p><em>|r| <= atol+rtol*|r0|</em></p>
<p>where <em>r0</em> is the initial residual and <em>|.|</em> is the energy norm. In fact</p>
<p><em>|r| = sqrt( bilinearform(Msolve(r),r))</em></p>
<p>For details on the preconditioned conjugate gradient method see the book:</p>
<p>“Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
C. Romine, and H. van der Vorst”.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
<li><strong>r</strong> (<em>any object supporting inplace add</em><em> (</em><em>x+=y</em><em>) </em><em>and scaling</em><em> (</em><em>x=scalar*y</em><em>)</em>) – initial residual <em>r=b-Ax</em>. <code class="docutils literal notranslate"><span class="pre">r</span></code> is altered.</li>
<li><strong>x</strong> (<em>any object supporting inplace add</em><em> (</em><em>x+=y</em><em>) </em><em>and scaling</em><em> (</em><em>x=scalar*y</em><em>)</em>) – an initial guess for the solution</li>
<li><strong>Aprod</strong> (function <code class="docutils literal notranslate"><span class="pre">Aprod(x)</span></code> where <code class="docutils literal notranslate"><span class="pre">x</span></code> is of the same object like
argument <code class="docutils literal notranslate"><span class="pre">x</span></code>. The returned object needs to be of the same type
like argument <code class="docutils literal notranslate"><span class="pre">r</span></code>.) – returns the value Ax</li>
<li><strong>Msolve</strong> (function <code class="docutils literal notranslate"><span class="pre">Msolve(r)</span></code> where <code class="docutils literal notranslate"><span class="pre">r</span></code> is of the same type like
argument <code class="docutils literal notranslate"><span class="pre">r</span></code>. The returned object needs to be of the same
type like argument <code class="docutils literal notranslate"><span class="pre">x</span></code>.) – solves Mx=r</li>
<li><strong>bilinearform</strong> (function <code class="docutils literal notranslate"><span class="pre">bilinearform(x,r)</span></code> where <code class="docutils literal notranslate"><span class="pre">x</span></code> is of the same
type like argument <code class="docutils literal notranslate"><span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">r</span></code> is. The returned value
is a <code class="docutils literal notranslate"><span class="pre">float</span></code>.) – inner product <code class="docutils literal notranslate"><span class="pre"><x,r></span></code></li>
<li><strong>atol</strong> (non-negative <code class="docutils literal notranslate"><span class="pre">float</span></code>) – absolute tolerance</li>
<li><strong>rtol</strong> (non-negative <code class="docutils literal notranslate"><span class="pre">float</span></code>) – relative tolerance</li>
<li><strong>iter_max</strong> (<code class="docutils literal notranslate"><span class="pre">int</span></code>) – maximum number of iteration steps</li>
</ul>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">the solution approximation and the corresponding residual</p>
</td>
</tr>
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first"><code class="docutils literal notranslate"><span class="pre">tuple</span></code></p>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Warning:</th><td class="field-body"><p class="first last"><code class="docutils literal notranslate"><span class="pre">r</span></code> and <code class="docutils literal notranslate"><span class="pre">x</span></code> are altered.</p>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="function">
<dt id="esys.escript.pdetools.TFQMR">
<code class="descclassname">esys.escript.pdetools.</code><code class="descname">TFQMR</code><span class="sig-paren">(</span><em>r</em>, <em>Aprod</em>, <em>x</em>, <em>bilinearform</em>, <em>atol=0</em>, <em>rtol=1e-08</em>, <em>iter_max=100</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.TFQMR" title="Permalink to this definition">¶</a></dt>
<dd><p>Solver for</p>
<p><em>Ax=b</em></p>
<p>with a general operator A (more details required!).
It uses the Transpose-Free Quasi-Minimal Residual method (TFQMR).</p>
<p>The iteration is terminated if</p>
<p><em>|r| <= atol+rtol*|r0|</em></p>
<p>where <em>r0</em> is the initial residual and <em>|.|</em> is the energy norm. In fact</p>
<p><em>|r| = sqrt( bilinearform(r,r))</em></p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
<li><strong>r</strong> (<em>any object supporting inplace add</em><em> (</em><em>x+=y</em><em>) </em><em>and scaling</em><em> (</em><em>x=scalar*y</em><em>)</em>) – initial residual <em>r=b-Ax</em>. <code class="docutils literal notranslate"><span class="pre">r</span></code> is altered.</li>
<li><strong>x</strong> (same like <code class="docutils literal notranslate"><span class="pre">r</span></code>) – an initial guess for the solution</li>
<li><strong>Aprod</strong> (function <code class="docutils literal notranslate"><span class="pre">Aprod(x)</span></code> where <code class="docutils literal notranslate"><span class="pre">x</span></code> is of the same object like
argument <code class="docutils literal notranslate"><span class="pre">x</span></code>. The returned object needs to be of the same type
like argument <code class="docutils literal notranslate"><span class="pre">r</span></code>.) – returns the value Ax</li>
<li><strong>bilinearform</strong> (function <code class="docutils literal notranslate"><span class="pre">bilinearform(x,r)</span></code> where <code class="docutils literal notranslate"><span class="pre">x</span></code> is of the same
type like argument <code class="docutils literal notranslate"><span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">r</span></code>. The returned value is
a <code class="docutils literal notranslate"><span class="pre">float</span></code>.) – inner product <code class="docutils literal notranslate"><span class="pre"><x,r></span></code></li>
<li><strong>atol</strong> (non-negative <code class="docutils literal notranslate"><span class="pre">float</span></code>) – absolute tolerance</li>
<li><strong>rtol</strong> (non-negative <code class="docutils literal notranslate"><span class="pre">float</span></code>) – relative tolerance</li>
<li><strong>iter_max</strong> (<code class="docutils literal notranslate"><span class="pre">int</span></code>) – maximum number of iteration steps</li>
</ul>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Return type:</th><td class="field-body"><p class="first"><code class="docutils literal notranslate"><span class="pre">tuple</span></code></p>
</td>
</tr>
<tr class="field-odd field"><th class="field-name">Warning:</th><td class="field-body"><p class="first last"><code class="docutils literal notranslate"><span class="pre">r</span></code> and <code class="docutils literal notranslate"><span class="pre">x</span></code> are altered.</p>
</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="function">
<dt id="esys.escript.pdetools.getInfLocator">
<code class="descclassname">esys.escript.pdetools.</code><code class="descname">getInfLocator</code><span class="sig-paren">(</span><em>arg</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.getInfLocator" title="Permalink to this definition">¶</a></dt>
<dd><p>Return a Locator for a point with the inf value over all arg.</p>
</dd></dl>
<dl class="function">
<dt id="esys.escript.pdetools.getSupLocator">
<code class="descclassname">esys.escript.pdetools.</code><code class="descname">getSupLocator</code><span class="sig-paren">(</span><em>arg</em><span class="sig-paren">)</span><a class="headerlink" href="#esys.escript.pdetools.getSupLocator" title="Permalink to this definition">¶</a></dt>
<dd><p>Return a Locator for a point with the sup value over all arg.</p>
</dd></dl>
</div>
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<h2>Others<a class="headerlink" href="#others" title="Permalink to this headline">¶</a></h2>
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