Trefftz co-chain calculus

We are concerned with a special class of discretizations of general linear transmission problems stated in the calculus of differential forms and posed on Rn. In the spirit of domain decomposition, we partition Rn = Ω ∪ Γ ∪ Ω+, Ω a bounded Lipschitz polyhedron, Γ := ∂Ω, and Ω+ unbounded. In Ω, we employ a mesh-based discrete co-chain model for differential forms, which includes schemes like finite element exterior calculus and discrete exterior calculus. In Ω+, we rely on a meshless Trefftz–Galerkin approach, i.e., we use special solutions of the homogeneous PDE as trial and test functions. Our key contribution is a unified way to couple the different discretizations across Γ. Based on the theory of discrete Hodge operators, we derive the resulting linear system of equations. As a concrete application, we discuss an eddy-current problem in frequency domain, for which we also give numerical results.