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. In the spirit of domain decomposition, we partition {mathbb {R}}^n=Omega cup Gamma cup Omega _{+}, Omega a bounded Lipschitz polyhedron, Gamma :=partial Omega , and Omega _{+} unbounded. In Omega , 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 _{+}, 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 . 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.
Highlights
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
The impact of illconditioning is limited due to the low number of degrees of freedom required for Trefftz methods, given their exponential convergence
The authors are not aware of any prior work addressing the coupling of Trefftz methods with a general framework for numerical schemes based on volume meshes, like co-chain calculus
Summary
The calculus of differential forms, known as exterior calculus, is a powerful tool for expressing a host of apparently different linear boundary value problems in a unifying way through differential forms; see [23, p. 265, Sections 1 and 2]. The calculus of differential forms, known as exterior calculus, is a powerful tool for expressing a host of apparently different linear boundary value problems in a unifying way through differential forms; see [23, p. We write Λk(Rn) for the space of differential forms of order k, 0 ≤ k ≤ n, n ∈ N∗, in Rn [8, p. Each member of the family of partial differential equations we are concerned with in this work can naturally be split into two sets of equations. The first involves the exterior derivative operator d and comprises the equilibrium equations du = (−1)l σ, dj = ψ − ψ0, in Rn,.
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