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

Chapter

## Abstract

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

## Keywords

Boundary Integral Equation Fredholm Operator Integral Kernel Approximation Space Wave Propagation Problem

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## Authors and Affiliations

1. 1.ETH ZürichZürichSwitzerland
2. 2.École PolytechniquePalaiseauFrance