Challenges and Applications of Boundary Element Domain Decomposition Methods

Boundary integral equation methods are well suited to represent the Dirichlet to Neumann maps which are required in the formulation of domain decomposition methods. Based on the symmetric representation of the local Steklov– Poincaré operators by a symmetric Galerkin boundary element method, we describe a stabilized variational formulation for the local Dirichlet to Neumann map. By a strong coupling of the Neumann data across the interfaces, we obtain a mixed variational formulation. For biorthogonal basis functions the resulting system is equivalent to nonredundant finite and boundary element tearing and interconnecting methods. We will also address several open questions, ideas and challenging tasks in the numerical analysis of boundary element domain decomposition methods, in the implementation of those algorithms, and their applications.