The coupling of Finite Element Methods (FEM) and Boundary Element Methods (BEM) is being applied to magnetic field problems since complex geometries or unbounded domains have to be considered. Moreover, if moving geometries are present, BEM can simplify calculations since locally no volume meshes are required. The coupling is being realized efficiently by Domain Decomposition (DD) methods allowing easily the application of parallel algorithms. Besides DD a second concept of parallelization (local parallelization) is being used in order to maintain parallel efficiency in the case of fixed, non-reducible BE-domains.
In our algorithm, multigrid methods are used as preconditioners for several types of subproblems arising from FEM or BEM within the DD framework. In particular operators with different mapping properties of integrating and differentiating type occur. It turns out that concepts originating from BEM are well suited for FEM and vice versa.
While 2D magnetic field problems lead to scalar equations, 3D problems involve usually a vector potential as primary unknown. Because of the infinite dimensional kernel of the differential operator, gauging conditions are imposed in order to guarantee unique solvability. We propose and analyze several types of gauging conditions. Moreover, we introduce the coupling of vector-valued FEM (vector potential) and scalar BEM (scalar potential) based on non-overlapping DD methods similar to the 2D case.
Acknowledgments: This work has been supported by the Austrian Science Fund - 'Fonds zur Förderung der wissenschaftlichen Forschung' - within SFB 013.