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# Tutorials, by Physics
```{highlight} none
```
Below we list examples which simulate particular physics problems so that users interested in a particular set of governing equations can easily locate a relevant example. Often PETSc will have several examples looking at the same physics using different numerical tools, such as different discretizations, meshing strategy, closure model, or parameter regime.
## Poisson
The Poisson equation
$$
-\Delta u = f
$$
is used to model electrostatics, steady-state diffusion, and other physical processes. Many PETSc examples solve this equation.
> Finite Difference
> : ```{eval-rst}
>
> :2D: `SNES example 5 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/snes/tutorials/ex5.c.html>`_
> :3D: `KSP example 45 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ksp/ksp/tutorials/ex45.c.html>`_
> ```
>
> Finite Element
> : ```{eval-rst}
>
> :2D: `SNES example 12 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/snes/tutorials/ex12.c.html>`_
> :3D: `SNES example 12 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/snes/tutorials/ex12.c.html>`_
> ```
## Elastostatics
The equation for elastostatics balances body forces against stresses in the body
$$
-\nabla\cdot \bm \sigma = \bm f
$$
where $\bm\sigma$ is the stress tensor. Linear, isotropic elasticity governing infinitesimal strains has the particular stress-strain relation
$$
-\nabla\cdot \left( \lambda I \operatorname{trace}(\bm\varepsilon) + 2\mu \bm\varepsilon \right) = \bm f
$$
where the strain tensor $\bm \varepsilon$ is given by
$$
\bm \varepsilon = \frac{1}{2} \left(\nabla \bm u + (\nabla \bm u)^T \right)
$$
where $\bm u$ is the infinitesimal displacement of the body. The resulting discretizations use PETSc's nonlinear solvers
Finite Element
: ```{eval-rst}
:2D: `SNES example 17 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/snes/tutorials/ex17.c.html>`_
:3D: `SNES example 17 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/snes/tutorials/ex17.c.html>`_
:3D: `SNES example 56 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/snes/tutorials/ex56.c.html>`_
```
If we allow finite strains in the body, we can express the stress-strain relation in terms of the Jacobian of the deformation gradient
$$
J = \mathrm{det}(F) = \mathrm{det}\left(\nabla u\right)
$$
and the right Cauchy-Green deformation tensor
$$
C = F^T F
$$
so that
$$
\frac{\mu}{2} \left( \mathrm{Tr}(C) - 3 \right) + J p + \frac{\kappa}{2} (J - 1) = 0
$$
In the example everything is expressed in terms of determinants and cofactors of $F$.
> Finite Element
> :
## Stokes
{doc}`physics/guide_to_stokes`
## Euler
Not yet developed
## Heat equation
The time-dependent heat equation
$$
\frac{\partial u}{\partial t} - \Delta u = f
$$
is used to model heat flow, time-dependent diffusion, and other physical processes.
> Finite Element
> : ```{eval-rst}
>
> :2D: `TS example 45 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ts/tutorials/ex45.c.html>`_
> :3D: `TS example 45 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ts/tutorials/ex45.c.html>`_
> ```
## Navier-Stokes
The time-dependent incompressible Navier-Stokes equations
$$
\begin{aligned}
\frac{\partial u}{\partial t} + u\cdot\nabla u - \nabla \cdot \left(\mu \left(\nabla u + \nabla u^T\right)\right) + \nabla p + f &= 0 \\
\nabla\cdot u &= 0 \end{aligned}
$$
are appropriate for flow of an incompressible fluid at low to moderate Reynolds number.
> Finite Element
> : ```{eval-rst}
>
> :2D: `TS example 46 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ts/tutorials/ex46.c.html>`_
> :3D: `TS example 46 <PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ts/tutorials/ex46.c.html>`_
> ```
|
