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<div id="version" align=right><b>petsc-3.7.5 2017-01-01</b></div>
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<A NAME="PCLSC"><H1>PCLSC</H1></A>
Preconditioning for Schur complements, based on Least Squares Commutators
<H3><FONT COLOR="#CC3333">Options Database Key</FONT></H3>
<DT><B>-pc_lsc_scale_diag </B> -Use the diagonal of A for scaling
<br>
<P>
<P>
<H3><FONT COLOR="#CC3333">Notes</FONT></H3>
This preconditioner will normally be used with PCFieldSplit to precondition the Schur complement, but
it can be used for any Schur complement system. Consider the Schur complement
<P>
<PRE>
S = A11 - A10 inv(A00) A01
</PRE>
<P>
<A HREF="../PC/PCLSC.html#PCLSC">PCLSC</A> currently doesn't do anything with A11, so let's assume it is 0. The idea is that a good approximation to
inv(S) is given by
<P>
<PRE>
inv(A10 A01) A10 A00 A01 inv(A10 A01)
</PRE>
<P>
The product A10 A01 can be computed for you, but you can provide it (this is
usually more efficient anyway). In the case of incompressible flow, A10 A10 is a Laplacian, call it L. The current
interface is to hang L and a preconditioning matrix Lp on the preconditioning matrix.
<P>
If you had called <A HREF="../KSP/KSPSetOperators.html#KSPSetOperators">KSPSetOperators</A>(ksp,S,Sp), S should have type MATSCHURCOMPLEMENT and Sp can be any type you
like (<A HREF="../PC/PCLSC.html#PCLSC">PCLSC</A> doesn't use it directly) but should have matrices composed with it, under the names "LSC_L" and "LSC_Lp".
For example, you might have setup code like this
<P>
<PRE>
<A HREF="../Sys/PetscObjectCompose.html#PetscObjectCompose">PetscObjectCompose</A>((<A HREF="../Sys/PetscObject.html#PetscObject">PetscObject</A>)Sp,"LSC_L",(<A HREF="../Sys/PetscObject.html#PetscObject">PetscObject</A>)L);
<A HREF="../Sys/PetscObjectCompose.html#PetscObjectCompose">PetscObjectCompose</A>((<A HREF="../Sys/PetscObject.html#PetscObject">PetscObject</A>)Sp,"LSC_Lp",(<A HREF="../Sys/PetscObject.html#PetscObject">PetscObject</A>)Lp);
</PRE>
<P>
And then your Jacobian assembly would look like
<P>
<PRE>
<A HREF="../Sys/PetscObjectQuery.html#PetscObjectQuery">PetscObjectQuery</A>((<A HREF="../Sys/PetscObject.html#PetscObject">PetscObject</A>)Sp,"LSC_L",(<A HREF="../Sys/PetscObject.html#PetscObject">PetscObject</A>*)&L);
<A HREF="../Sys/PetscObjectQuery.html#PetscObjectQuery">PetscObjectQuery</A>((<A HREF="../Sys/PetscObject.html#PetscObject">PetscObject</A>)Sp,"LSC_Lp",(<A HREF="../Sys/PetscObject.html#PetscObject">PetscObject</A>*)&Lp);
if (L) { assembly L }
if (Lp) { assemble Lp }
</PRE>
<P>
With this, you should be able to choose LSC preconditioning, using e.g. ML's algebraic multigrid to solve with L
<P>
<PRE>
-fieldsplit_1_pc_type lsc -fieldsplit_1_lsc_pc_type ml
</PRE>
<P>
Since we do not use the values in Sp, you can still put an assembled matrix there to use normal preconditioners.
<P>
<H3><FONT COLOR="#CC3333">References</FONT></H3>
<TABLE border="0" cellpadding="0" cellspacing="0">
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>1. </B></TD><TD>- Elman, Howle, Shadid, Shuttleworth, and Tuminaro, Block preconditioners based on approximate commutators, 2006.
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>2. </B></TD><TD>- Silvester, Elman, Kay, Wathen, Efficient preconditioning of the linearized Navier Stokes equations for incompressible flow, 2001.
</TD></TR></TABLE>
<P>
<P>
<H3><FONT COLOR="#CC3333">See Also</FONT></H3>
<A HREF="../PC/PCCreate.html#PCCreate">PCCreate</A>(), <A HREF="../PC/PCSetType.html#PCSetType">PCSetType</A>(), <A HREF="../PC/PCType.html#PCType">PCType</A> (for list of available types), <A HREF="../PC/PC.html#PC">PC</A>, Block_Preconditioners, <A HREF="../PC/PCFIELDSPLIT.html#PCFIELDSPLIT">PCFIELDSPLIT</A>,
<BR><A HREF="../PC/PCFieldSplitGetSubKSP.html#PCFieldSplitGetSubKSP">PCFieldSplitGetSubKSP</A>(), <A HREF="../PC/PCFieldSplitSetFields.html#PCFieldSplitSetFields">PCFieldSplitSetFields</A>(), <A HREF="../PC/PCFieldSplitSetType.html#PCFieldSplitSetType">PCFieldSplitSetType</A>(), <A HREF="../PC/PCFieldSplitSetIS.html#PCFieldSplitSetIS">PCFieldSplitSetIS</A>(), <A HREF="../PC/PCFieldSplitSetSchurPre.html#PCFieldSplitSetSchurPre">PCFieldSplitSetSchurPre</A>(),
<A HREF="../KSP/MatCreateSchurComplement.html#MatCreateSchurComplement">MatCreateSchurComplement</A>()
<P><B><P><B><FONT COLOR="#CC3333">Level:</FONT></B>intermediate
<BR><FONT COLOR="#CC3333">Location:</FONT></B><A HREF="../../../src/ksp/pc/impls/lsc/lsc.c.html#PCLSC">src/ksp/pc/impls/lsc/lsc.c</A>
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