Coupling finite elements and auxiliary sources

Abstract We consider second-order scalar elliptic boundary value problems on unbounded domains, which model, for instance, electrostatic fields. We propose a discretization that relies on a Trefftz approximation by multipole auxiliary sources in some parts of the domain and on standard mesh-based primal Lagrangian finite elements in other parts. Several approaches are developed and, based on variational saddle point theory, rigorously analyzed to couple both discretizations across the common interface: 1. Least-squares-based coupling using techniques from PDE-constrained optimization. 2. Coupling through Dirichlet-to-Neumann operators. 3. Three-field variational formulation in the spirit of mortar finite element methods. We compare these approaches in a series of numerical experiments.