First-Kind Boundary Integral Equations for the Hodge--Helmholtz Operator

Abstract

We adapt the variational approach to the analysis of first-kind boundary integral equations associated with strongly elliptic partial differential operators from [M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), pp. 613-626.] to the (scaled) Hodge-Helmholtz equation curl curl u − η∇div u − κ 2 u = 0, η > 0, Im κ 2 ≥ 0, on Lipschitz domains in 3D Euclidean space, supplemented with natural complementary boundary conditions, which, however, fail to bring about strong ellipticity. Nevertheless, a boundary integral representation formula can be found, from which we can derive boundary integral operators. They induce bounded and coercive sesqui-linear forms in the natural energy trace spaces for the Hodge-Helmholtz equation. We can establish precise conditions on η, κ that guarantee unique solvability of the two first-kind boundary integral equations associated with the natural boundary value problems for the Hodge-Helmholtz equations. Particular attention will be given to the case κ = 0.

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

“…Straightforward integration by parts, elaborated, for instance, in Claeys and Hiptmair, Sect. 3 (19) and Kress,, Lemma 3.2 reveals that the fundamental symmetric bilinear form induced by is It is a more subtle matter to decide about meaningful choices for function spaces, on which to pose variational problems for .…”

“…This will define boundary conditions and point to relevant trace operators. Since we rely on both to state pertinent boundary integral equations, we summarize Claeys and Hiptmair, Sect. 3 in this section.…”

“…This article builds upon the work of Kress and our theoretical investigations in Claeys and Hiptmair, Sect. 6 into boundary value problems for the Hodge‐Helmholtz operator , , and associated first‐kind boundary integral equations.…”

“…6 into boundary value problems for the Hodge‐Helmholtz operator , , and associated first‐kind boundary integral equations. To establish the setting, we will review and extend the results of Claeys and Hiptmair, Sects. 3 and 4 in Sections and .…”

“…Remark When enforcing vanishing divergence of solutions of by choosing suitable boundary conditions, the Hodge‐Laplacian offers a way to express the zero‐frequency limit of Maxwell equations in frequency domain, cf the introduction of Claeys and Hiptmair or Hazard and Lenoir . Integral equations for the time‐harmonic Maxwell equations are well established and usually comprise the so‐called electric‐field and magnetic‐field boundary integral operators (BIOs) ,.…”