# A Gallery of finite element solvers

# A Gallery of finite element solvers¶

The goal of this chapter is to demonstrate how a range of important PDEs from science and engineering can be quickly solved with a few lines of DOLFINx code. We will start with the heat equation, then continue with the non-linear Poisson equation, the equations for linear elasticity, the Navier-Stokes equations, and finally look at how to solve systems of nonlinear advection-diffusion-reaction equations. These problems illustrate how to solve time-dependent problems, non-linear problems, vector-values problems and systems of PDEs. For each problem, we derive the variational formulation and express the problem in Python in a way that closely resembels the mathematics.