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First-Kind Galerkin Boundary Element Methods for the Hodge-Laplacian in Three Dimensions

by X. Claeys and R. Hiptmair

(Report number 2018-35)

Abstract
Boundary value problems for the Euclidean Hodge-Laplacian in three dimensions \(\boldsymbol{\Delta}_{\mathrm{HL}} := \mathbf{curl}\,\mathbf{curl} - \mathbf{grad}\,\mathrm{div}\) lead to variational formulations set in subspaces of \(\boldsymbol{H}(\mathbf{curl},\Omega)\cap\boldsymbol{H}(\mathrm{div},\Omega)\), \(\Omega\subset\mathbb{R}^{3}\) a bounded Lipschitz domain. Via a representation formula and Calder\'on identities we derive corresponding first-kind boundary integral equations set in trace spaces of \(H^{1}(\Omega)\), \(\boldsymbol{H}(\mathbf{curl},\Omega)\), and \(\boldsymbol{H}(\mathrm{div},\Omega)\). They give rise to saddle-point variational formulations and feature kernels whose dimensions are linked to fundamental topological invariants of \(\Omega\). Kernels of the same dimensions also arise for the linear systems generated by low-order conforming Galerkin boundary element (BE) discretization. On their complements, we can prove stability of the discretized problems, nevertheless. We prove that discretization does not affect the dimensions of the kernels and also illustrate this fact by numerical tests.

Keywords: Hodge-Laplacian, representation formula, Calder\'on indentities, first-kind boundary integral equations, harmonic vector fields, saddle-point problems, boundary element method (BEM)

@Techreport{CH18_789,
  author = {X. Claeys and R. Hiptmair},
  title = {First-Kind Galerkin Boundary Element Methods for the Hodge-Laplacian in Three Dimensions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2018-35},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2018/2018-35.pdf },
  year = {2018}
}

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