Boundary value problems for the Euclidean Hodge‐Laplacian in three dimensions $-{\mathrm{\Delta }}_{\text{HL}}:=\mathbf{curl}\mathbf{curl}-\mathbf{grad}\text{div}$ lead to variational formulations set in subspaces of $\mathit{H}(\mathbf{curl},\mathrm{\Omega })\cap \mathit{H}(\text{div},\mathrm{\Omega })$, $\mathrm{\Omega }\subset {\mathbb{R}}^3$ a bounded Lipschitz domain. Via a representation formula and Calderón identities, we derive corresponding first‐kind boundary integral equations set in trace spaces of $H^1(\mathrm{\Omega })$, $\mathit{H}(\mathbf{curl},\mathrm{\Omega })$, and $\mathit{H}(\text{div},\mathrm{\Omega })$. They give rise to saddle‐point variational formulations and feature kernels whose dimensions are linked to fundamental topological invariants of $\mathrm{\Omega }$. 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.

中文翻译：

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三维Hodge-Laplacian的第一类Galerkin边界元方法

欧氏Hodge-Laplacian的三维边值问题 $-{\mathrm{\Delta }}_{\text{HL}}：=\mathbf{卷曲}\mathbf{卷曲}-\mathbf{毕业}\text{div}$ 导致在的子空间中设置变式 $\mathit{H}（\mathbf{卷曲}，\mathrm{\Omega }）\cap \mathit{H}（\text{div}，\mathrm{\Omega }）$， $\mathrm{\Omega }\subset {\mathbb{[R}}^3$有界的Lipschitz域。通过一个表示公式和Calderón恒等式，我们得出在的迹线空间中设置的相应的第一类边界积分方程$H^{\mathrm{1个}}（\mathrm{\Omega }）$， $\mathit{H}（\mathbf{卷曲}，\mathrm{\Omega }）$和 $\mathit{H}（\text{div}，\mathrm{\Omega }）$。它们产生了鞍点变分公式和特征核，其维数与图的基本拓扑不变量相关$\mathrm{\Omega }$。由低阶一致性Galerkin（BE）离散化生成的线性系统也会出现相同尺寸的核。在它们的补充上，我们仍然可以证明离散问题的稳定性。我们证明离散化不会影响内核的尺寸，并且还通过数值测试说明了这一事实。