First-kind Galerkin boundary element methods for the Hodge-Laplacian in three dimensions

Boundary value problems for the Euclidean Hodge-Laplacian in three dimensions (Formula presented.) lead to variational formulations set in subspaces of (Formula presented.), (Formula presented.) a bounded Lipschitz domain. Via a representation formula and Calderón identities, we derive corresponding first-kind boundary integral equations set in trace spaces of (Formula presented.), (Formula presented.), and (Formula presented.). They give rise to saddle-point variational formulations and feature kernels whose dimensions are linked to fundamental topological invariants of (Formula presented.). 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.