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

Xavier Claeys (1, 2) , Ralf Hiptmair (3)

1 ALPINES - Algorithms and parallel tools for integrated numerical simulations
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
3 SAM - Seminar for Applied Mathematics [Zürich]

Boundary value problems for the Euclidean Hodge-Laplacian in three dimensions $−∆ HL := curl curl−grad$ div lead to variational formulations set in subspaces of $H(curl, Ω)∩ H(div, Ω)$, $Ω ⊂ R 3$ a bounded Lipschitz domain. Via a representation formula and Calderón identities we derive corresponding first-kind boundary integral equations set in trace spaces of $H 1 (Ω)$, $H(curl, Ω$), and $H(div, Ω)$. They give rise to saddle-point variational formulations and feature kernels whose dimensions are linked to fundamental topological invariants of $Ω$. 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.

Mots clés: First-kind boundary integral equations; Calderón indentities; Representation formula; Saddle-point problems; Harmonic vector fields; Hodge-Laplacian; Boundary element method (BEM)

Domaines: Analyse numérique [math.NA]

Xavier Claeys, Ralf Hiptmair. First-Kind Galerkin Boundary Element Methods for the Hodge-Laplacian in Three Dimensions. Mathematical Methods in the Applied Sciences, In press. ⟨hal-02425493⟩