# Guide to PETSc Tutorial Examples, 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

## 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.

- Finite Element

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 itself, everything can be expressed in terms of determinants and cofactors of \(F\).

- Finite Element

## Stokes¶

The Stokes equations

describe slow flow of an incompressible fluid with velocity \(u\), pressure \(p\), and body force \(f\).

- Finite Element
See Guide to the Stokes Equations using Finite Elements in PETSc for more.

## Euler¶

## Heat equation¶

The heat equation

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

- Finite Element

- 2D
- 3D