Boundary integral exterior calculus

  • Erick Schulz

    Plexim GmbH, Zürich, Switzerland
  • Ralf Hiptmair

    ETH Zurich, Zürich, Switzerland
  • Stefan Kurz

    ETH Zurich, Zürich, Switzerland
Boundary integral exterior calculus cover
Download PDF

A subscription is required to access this article.

Abstract

We report a surprising and deep structural property of boundary integral operators occurring in first-kind boundary integral equations associated with Hodge–Dirac and Hodge–Laplace operators for de Rham Hilbert complexes on a bounded domain in a Riemannian manifold. We show that, as regards their induced bilinear forms, those boundary integral operators are Hodge–Dirac and Hodge–Laplace operators in the weak sense, this time set in a trace de Rham Hilbert complex on the boundary whose underlying spaces of differential forms are equipped with non-local inner products defined through layer potentials. On the way to this main result we conduct a thorough analysis of layer potentials in operator-induced trace spaces and derive representation formulas.

Cite this article

Erick Schulz, Ralf Hiptmair, Stefan Kurz, Boundary integral exterior calculus. J. Eur. Math. Soc. (2025), published online first

DOI 10.4171/JEMS/1621