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First-kind Galerkin boundary element methods for the Hodge-Laplacian in three dimensions. (English) Zbl 1446.31002

Summary: Boundary value problems for the Euclidean Hodge-Laplacian in three dimensions \(-\Delta_{\text{HL}} := \mathbf{curl} \, \mathbf{curl} - \mathbf{grad} \operatorname{ div}\) lead to variational formulations set in subspaces of \(\boldsymbol{H}(\mathbf{curl}, \Omega) \cap \boldsymbol{H}(\operatorname{div}, \Omega)\), \(\Omega \subset \mathbb{R}^3\) a bounded Lipschitz domain. Via a representation formula and Calderón identities, we derive corresponding first-kind boundary integral equations set in trace spaces of \(H^1(\Omega), \boldsymbol{H}(\mathbf{curl}, \Omega)\), and \(\boldsymbol{H}(\operatorname{div}, \Omega)\). 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 (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.

MSC:

31A10 Integral representations, integral operators, integral equations methods in two dimensions
45A05 Linear integral equations
65N38 Boundary element methods for boundary value problems involving PDEs
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