Trefftz co-chain calculus

Abstract

We are concerned with a special class of discretizations of general linear transmission problems stated in the calculus of differential forms and posed on $${\mathbb {R}}^n$$ R n . In the spirit of domain decomposition, we partition $${\mathbb {R}}^n=\Omega \cup \Gamma \cup \Omega _{+}$$ R n = Ω ∪ Γ ∪ Ω + , $$\Omega $$ Ω a bounded Lipschitz polyhedron, $$\Gamma :=\partial \Omega $$ Γ : = ∂ Ω , and $$\Omega _{+}$$ Ω + unbounded. In $$\Omega $$ 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 $$\Omega _{+}$$ 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 $$\Gamma $$ 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.