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

Abstract

Consider a flat smooth manifold $\Gamma_m \subset \mathbb{R}^3$ of codimension one with Lipschitz boundary ∂Γ_m 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 Γ_m. If [·]Γ_m denotes the jump across Γ_m, clearly $[\gamma D u]_{\Gamma_m} = 0.$. Thus, solutions over Ω can be built [Mc00] via the single-layer potential Ψ^k _SL, i.e.,

$$u({\bf x}) = -{\bf \Psi}^k_{SL} ([\gamma N u]_{\Gamma_m}(\bf x) \quad {\rm for} \ {\bf x}\in \Omega,$$

(19.1)

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

$$G_k({\bf z}) = \frac{1}{4\pi}\frac{\exp (ik|{\bf z}|)}{|{\bf z}|} \qquad {\rm for}\; k \in \mathbb{R},$$

(19.2)

being the associated fundamental solution of the differential equation.