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Boundary Integral Exterior Calculus

by E. Schulz and R. Hiptmair and S. Kurz

(Report number 2022-36)

Abstract
We report a surprising and deep structural property of first-kind boundary integral operators for Hodge--Dirac and Hodge--Laplace operators associated with de Rham Hilbert complexes on a bounded domain $\Omega$ in a Riemannian manifold. We show that from a variational perspective, those first-kind boundary integral operators are Hodge--Dirac and Hodge--Laplace operators as well, this time set in a trace de Rham Hilbert complex on the boundary $\partial\Omega$ 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.

Keywords: Hodge-Dirac operator, Hodge-Laplace operator, Hodge-Yukawa, representation formula, boundary integral operators, compact manifolds, structure-preserving

@Techreport{SHK22_1024,
  author = {E. Schulz and R. Hiptmair and S. Kurz},
  title = {Boundary Integral Exterior Calculus},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-36},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-36.pdf },
  year = {2022}
}

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