SpaceTimeFEM_2023-2024

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SpaceTimeFEM_2023-2024

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

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# 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

SpaceTimeFEM_2023-2024

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