Boundary Hybrid Galerkin Method for Elliptic and Wave Propagation Problems in ℝ3 over Planar Structures

Consider a flat smooth manifold $$\Gamma_m \subset \mathbb{R}^3$$ of codimension one with Lipschitz boundary ∂Γm and large aspect ratios such as the one depicted in Figure 19.1. Let the associated unbounded domain $$\Omega := \mathbb{R}^3 \backslash \bar{\Gamma}_m$$ be isotropic and homogeneous for the moment. We seek solutions $$u \in H^1_{loc}(\Omega)$$ of the Laplace and Helmholtz equations when a Dirichlet condition gD is applied on Γm such that $$\gamma^+_D u |_{\Gamma_m} = \gamma^-_D u|_{\Gamma_m} = gD \in H^{1/2}(\Gamma_m),$$ where γ± D are the Dirichlet trace operators from either side of Γm. If [·]Γm denotes the jump across Γm, clearly $$[\gamma D u]_{\Gamma_m} = 0.$$ . Thus, solutions over Ω can be built [Mc00] via the single-layer potential Ψk SL, i.e., 19.1 $$u({\bf x}) = -{\bf \Psi}^k_{SL} ([\gamma N u]_{\Gamma_m}(\bf x) \quad {\rm for} \ {\bf x}\in \Omega,$$ where $${\bf \Psi}^k_{SL}(\varphi)(\bf x): = \int_{\Gamma_m} G_k ({\bf x}-{\bf y})\varphi({\bf y})d{\bf y}\qquad{\rm for} \ {\bf x}\in \Omega,$$ γN is the Neumann trace operator, and the integral kernel G k takes the form 19.2 $$G_k({\bf z}) = \frac{1}{4\pi}\frac{\exp (ik|{\bf z}|)}{|{\bf z}|} \qquad {\rm for}\; k \in \mathbb{R},$$ being the associated fundamental solution of the differential equation.