Numerical Methods for Partial Differential Equations
Partial differential equations are used in science and engineering to describe phenomena like electric fields, waves, structural deformations, or heat transport. In practice these equations have to be solved by numerical methods, and the development of efficient algorithms for partial differential equations is an important subject in both numerical analysis and numerical linear algebra. This lecture focuses on the discretization of these equations, i.e., the approximation of the solution in a finite-dimensional space, while the lecture "Iterative methods for large linear systems" treats efficient solvers for the resulting systems of equations. It covers the following topics:
- finite difference discretization of elliptic equations (Poisson's equation, consistency, stability, convergence),
- finite difference discretization of parabolic equations (heat equation, stable time-stepping schemes, convergence),
- finite difference discretization of hyperbolic equations (wave equation, stable time-stepping schemes, conservation of energy, convergence),
- variational problems and the finite element method (Sobolev spaces, Riesz' theorem, Lax-Milgram, Friedrichs' inequality, Galerkin's method, Cea's lemma),
- finite element method (triangulation, nodal basis functions, stiffness matrix, Bramble-Hilbert lemma, Sobolev's lemma, interpolation error).
The exercises accompanying the lecture focus both on extensions of the mathematical theory and the implementation and application of the presented techniques.