After successful completion of the course, students are able to formulate certain partial differential equations as equivalent integral equations. They can discretize these equations via a Galerkin method and understand the underlying mathematical theory.
Classical numerical algorithms, like the finite element method, is based on decomposing the underlying domain of interest and solve a discretized version of the partial differential equation on these pieces. An alternative approach is to reformulate the problem as an equivalent integral equation on the boundary of the domain using the fundamental solution. Once reformulated, the problem can then be discretized using a Galerkin method.
This approach has multiple advantages:
1) reduction of dimension: instead of a 2d domain, only a 1d curve has to be discretized
2) unbounded domains can also be considered
3) better convergence rates compared to finite elment methods
This lecture introduces the theory of boundary element methods (BEM). It lays the mathematical foundation as well as deals with more practical aspects of the method.
Topics include:
- derivation of the representation formula and integral equations
- a priori convergence theory
- assembly of the stiffness matrix and numerical quadrature for singular integrals
- matrix compression techniques, e.g. using H-matrices
Numerical analyis, partial differential equations.
Experience with finite element methods is helpful.
