SpaceTimeFEM_2023-2024

Exercises for the course "Space-time methods" at the Leibniz University Hannover in the winter semester 2023/2024

Exercise 1

Introduction to:

• Python
• SymPy
• FEniCS
• git

Homework for Exercise 1:

Solve heat equation with space-time FEM in FEniCS.

$$ \partial_t u - \partial_{xx} u = f $$

Exercise 2

Solve ODE

$$ \begin{align} \partial_t u &= \lambda u, \\ u(0) &= u_0, \end{align} $$

using SymPy and NumPy.

Homework for Exercise 2:

Use same methodology for

$$ \begin{align} \partial_t v &= -u, \\ \partial_t u &= v, \\ u(0) &= 1, \\ v(0) &= 0. \end{align} $$

Exercise 3

Solve heat equation (from the Homework to Exercise 1) with tensor-product space-time FEM. Implementation in FEniCS.

Homework for Exercise 3:

Try to find ways to speed up the code from Exercise 3.

Exercise 4

1. Solve the nonlinear heat equation

$$ \partial_t u - \partial_{xx} u + u^2 = f $$

with tensor-product space-time FEM in FEniCS. Time integration is performed with SymPy and the spatial FEM (incl. automatic differentiation) is done in FEniCS.

2. Solve the time-dependent Navier-Stokes equations

$$ \partial_t v - \nabla_x \cdot \sigma + (v \cdot \nabla_x)v = 0, $$

$$ \nabla_x \cdot v = 0, $$

$$ \sigma = \sigma \begin{pmatrix} v \\ p \end{pmatrix} = -pI + \nu \nabla_x v, $$

with tensor-product space-time FEM in FEniCS.

3. Presentation of topics for group projects