An abstract domain decomposition framework for heterogeneous coupling problems: stability, approximation and applications

With the advent of exascale high performance computing there is an increasing interest in fast methods for the solution of the linear systems that appear after discretization of partial differential equations. In particular heterogeneous problems where systems are coupled, either over an interface or on a bulk overlap, may be challenging because of different properties of the coupled physical systems. In this talk we will take a recent computational project on the electrostatic potential in molecular solvation as starting point for the discussion. For this problem a heterogeneous Boltzmann-Poisson equation close to the molecule must be coupled to a homogeneous Poisson equation in the far field, which prompts the coupling of finite element and boundary element methods. We will then introduce an abstract theoretical framework for domain decomposition that is agnostic to the numerical methods used for the different subsolvers. This naturally leads to a set of assumptions which must be satisfied by the bulk and coupling operators for the well-posedness of the reduced system, where the bulk fields have been eliminated. Having established the theory on the continuous level we consider discretization of the systems and discuss the stability, stabilisation and preconditioning of the formulation. Finally we will discuss some applications to different heterogeneous coupling problems such as FEM-BEM coupling, the Arlequin method and mixed dimensional problems.