After successful completion of the course, students are able to numerically solve linear equation systems directly and iteratively. Students know to set up boundary and initial boundary value problems in principle and in particular for Maxwell’s equations and to solve them approximately by means of the finite difference method to some extent. Furthermore, students are essentially able to derive the weak formulation of simple boundary value problems and to find an approximate solution either by implementing an own computer code of the finite element method for instance in Python or by using Netgen/NGSolve (Open Source Software) to model and simulate problems in electrical engineering and to carry out a validity check.

Linear equation systems, introduction to partial differential equations, their classification and some essential properties, establishing the initial boundary and boundary value problems based on Maxwell’s equations, discussion of the practical relevance, approximate solution with the finite difference method, method of weighted residuals, idea of the finite element method, derivation of the weak formulation, assembling of the finite element equation systems using hat functions, weak formulations based on a scalar potential and on a vector potential in the context of the Maxwell’s equations, construction of finite element bases for the Sobolev spaces H^1 and H(curl).

Some simple algorithms are to be implemented in the associated exercises. Small but relevant problems in electrical engineering will be solved using Netgen/NGSolve. To this end partly prepared examples in Python will be provided, which have to be completed or extended.

In small groups of 2 or 3 students mathematical exercises have to be solved, simple algorithms have to be implemented and analyzed and simulation exercises have to be carried out and the results discussed. One protocol has to be prepared together by the group.

Skript is available

Calculus, Ordinary Differential Equations and Linear Algebra