The students learn how to apply their theoretical knowledge, for instance from their lectures on functional analysis, for the mathematical and numerical treatment of partial differential equations (PDEs). The boundary element method (BEM) is a numerical scheme for the solution of ellliptic PDEs, which is in some sense superior to the well-established finite element method (FEM). For instance, the BEM allows for the treatment of unbounded domains and leads to higher convergence rates when compared with the lowest order FEM. The lecture introduces the functional analytic framework of the BEM as well as the consequences for a numerical treatment. In this sense an alternate title of the lecture could have been "Applied Functional Analysis".

The contents of the lecture read as follows: (1) Strong and weak formulation of elliptic PDEs (2) Sobolev spaces on domains and boundaries (3) Equivalent integral formulation of elliptic PDEs (4) Properties of the involved integral operators (in particular, convolution operators) (5) Galerkin schemes (6) Boundary element method: ansatz and a priori error estimates (7) Numerical and theoretical comparison of FEM and BEM (8) Applications of BEM

There are theoretical exercises (1 hour) which accompany the lecture. Practical exercises (e.g., the implementation of BEM) can be done within a 5- or 10-hour lab, which can be the basis for a bachelor or diploma thesis. By wish of the participants (or even by necessity), this course can be held in English.

Lecture notes for this course are available. Es wird sukzessive in der Vorlesung (kostenlos) an die Teilnehmer ausgegeben.