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

  • C. Jerez-Hanckes
  • J.-C. Nédélec
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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2010

Authors and Affiliations

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

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