Most physical processes are modeled by partial differential equations (PDEs) and most of these equations cannot be solved explicitly. That is where numerical methods come into play. This course will introduce the methods of finite differences and finite elements for both stationary and time dependent problems. We will discuss theoretical aspects like consistency, stability, convergence, and error analysis of the given method as well as practical aspects of implementation. We will concentrate on elliptic and parabolic PDEs

and only briefly touch on hyperbolic and dispersive problems if time permits.

Prerequisites: linear algebra is necessary while the basics in PDEs and functional analysis are helpful

# Exercise sheets:

**SHEET 1** - FD codes (Matlab): Poisson equ., stationary heat equ.

**SHEET 2** - FD codes (Matlab): Poisson equ. in 2D (Prob. 2), 5-point stencil approximation of the 2D Laplacian (Prob. 2a), 4th order approximation of the 2D Laplacian (Prob. 2b), Poisson equ. in 2D in polar coordinates (Prob. 3c), 2nd order approx. of the 2D Laplacian in polar coord. (Prob. 3c)

**SHEET 3**- FD codes (Matlab): eigenvalues of the matrix corresp. to the 5-point stencil for the second derivative in 2D, the matrix 5-point corresp. to the 5-point stencil for the second derivative in 2D

**SHEET 4** - FD codes (Matlab): Laplace equ. on a disk with a rerentrant corner (in polar coordinates) (Prob. 3b), 2nd order approx. of the 2D Laplacian in polar coord. for the reentrant corner problem(Prob. 3b)

**SHEET 6** - FEM codes (Matlab): quadratic FEM for the 1D Poisson equ. (Prob. 3), elliptic problem in 2D solved via the PDE toolbox (Prob. 4)

**SHEET 7** - FEM codes (Matlab): linear FEM for the 1D Poisson equ. (Prob. 1), load vector for f(x)=1, load vector for f(x)=x^2

# References

- R.J. LeVeque, Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems, Society for Industrial and Applied Mathematics.
- D. Braess, Finite elements: theory, fast solvers, and applications in solid mechanic, Cambridge Univ. Press.
- P. Knabner and L. Angermann, Numerical methods for elliptic and parabolic partial differential equations, Springer.
- S. Larsson and V. Thomee, Partial Differential Equations with Numerical Methods, Springer.
- A. Quarteroni, R. Sacco, and F. Saleri, Numerical mathematics, Springer.