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 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.

H 1 -conforming finite element cochain complexes and commuting quasi-interpolation operators on Cartesian meshes

A finite element cochain complex on Cartesian meshes of any dimension based on the H 1 -inner product is introduced. It yields H 1 -conforming finite element spaces with exterior derivatives in H 1 . We use a tensor product construction to obtain L 2 -stable projectors into these spaces which commute with the exterior derivative. The finite element complex is generalized to a family of arbitrary order.