Trefftz co-chain calculus

Z Angew Math Phys. 2022;73(1):43. doi: 10.1007/s00033-021-01671-y. Epub 2022 Jan 25.

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 R n . In the spirit of domain decomposition, we partition R n = 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.

Keywords: Co-chain calculus; Discrete exterior calculus; Finite element exterior calculus; Trefftz method.