Section: New Results

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

Boundary value problems for the Euclidean Hodge-Laplacian in three dimension $-{\Delta }_{HL}=\mathrm{𝐜𝐮𝐫𝐥}\mathrm{𝐜𝐮𝐫𝐥}-\mathrm{𝐠𝐫𝐚𝐝}\mathrm{div}$ lead to variational formulations set in subspaces of $𝐇(\mathrm{𝐜𝐮𝐫𝐥},\Omega )\cap 𝐇(\mathrm{div},\Omega )$, $\Omega \subset {ℝ}^3$ a bounded Lipschitz domain. Via a representation formula and Calderón identities we derive corresponding first-kind boundary integral equations set in trace spaces of $H^1\left(\Omega \right)$, $𝐇(\mathrm{𝐜𝐮𝐫𝐥},\Omega )$, and $𝐇(\mathrm{div},\Omega )$. 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.