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First‐kind Galerkin boundary element methods for the Hodge‐Laplacian in three dimensions
Author(s) -
Claeys Xavier,
Hiptmair Ralf
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6203
Subject(s) - mathematics , discretization , galerkin method , lipschitz domain , mathematical analysis , boundary (topology) , hodge dual , linear subspace , bounded function , laplace operator , lipschitz continuity , domain (mathematical analysis) , boundary value problem , pure mathematics , hodge theory , finite element method , cohomology , physics , thermodynamics
Boundary value problems for the Euclidean Hodge‐Laplacian in three dimensions − Δ HL : = curl curl − grad div lead to variational formulations set in subspaces of H ( curl , Ω ) ∩ H ( div , Ω ) , Ω ⊂ R 3a bounded Lipschitz domain. Via a representation formula and Calderón identities, we derive corresponding first‐kind boundary integral equations set in trace spaces ofH 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 (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.