Title: Boundary Element Exterior Calculus

Abstract: We consider first-kind boundary integral equations (BIEs) arising from boundary value problems for the Hodge-Laplacians and the Dirac operator associated with the de Rham complex on a bounded Lipschitz domain in Euclidean space. We discovered [1] that the variational formulations of these BIEs are those of the Hodge-Laplacians and Dirac operator belonging to the so-called trace de Rham complex [2], which is a Hilbert complex of differential forms on the boundary of the domain with non-local inner products. Hence the conforming boundary-element Galerkin discretization of these BIEs can rely on the framework of finite-element exterior calculus with the new twist of a non-local inner product. In particular, once we establish an h-uniform discrete Poincare inequality involving non-local norms, h-uniform stability of the boundary element method is settled.

[1] E. Schulz, R. Hiptmair, and S. Kurz, Boundary integral exterior calculus, Tech. Rep. 2022-36, Seminar for Applied Mathematics, ETH Zürich, 2022

[2] R. Hiptmair, D. Pauly, and E. Schulz, Traces for Hilbert complexes, J. Funct. Anal., 284 (2023)