# Tutorials, by Physics#

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

is used to model electrostatics, steady-state diffusion, and other physical processes. Many PETSc examples solve this equation.

- Finite Difference

- 2D:
- 3D:
- Finite Element

- 2D:
- 3D:

## Elastostatics#

The equation for elastostatics balances body forces against stresses in the body

where \(\bm\sigma\) is the stress tensor. Linear, isotropic elasticity governing infinitesimal strains has the particular stress-strain relation

where the strain tensor \(\bm \varepsilon\) is given by

where \(\bm u\) is the infinitesimal displacement of the body. The resulting discretizations use PETSc’s nonlinear solvers

- Finite Element
- 2D:
- 3D:
- 3D:

If we allow finite strains in the body, we can express the stress-strain relation in terms of the Jacobian of the deformation gradient

and the right Cauchy-Green deformation tensor

so that

In the example everything is expressed in terms of determinants and cofactors of \(F\).

- Finite Element

- 3D:

## Stokes#

## Euler#

Not yet developed

## Heat equation#

The time-dependent heat equation

is used to model heat flow, time-dependent diffusion, and other physical processes.

- Finite Element

- 2D:
- 3D: