Boundary integral equations for screen problems in \({\mathbb{R}}^ 3\). (English) Zbl 0653.35016

Stephan, Ernst P.

Integral Equations Oper. Theory 10, 236-257 (1987).

We present a new solution procedure for Helmholtz and Laplacian Neumann screen or Dirichlet screen problems in \({\mathbb{R}}^ 3\) via boundary integral equations of the first kind having as unknown the jump of the field or of its normal derivative, respectively, across the screen S. Under the assumption of local finite energy we show the equivalence of the integral equations and the original boundary value problems. Via the Wiener-Hopf method in the halfspace, localization and the calculus of pseudodifferential operators we derive existence, uniqueness and regularity results for the solution of our boundary integral equations together with its explicit behavior near the edge of the screen. We give Galerkin schemes based on our integral equations on S and obtain high convergence rates by using special singular elements besides regular splines as test and trial functions.

MSC:

35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25	Boundary value problems for second-order elliptic equations
35A05	General existence and uniqueness theorems (PDE) (MSC2000)
35B65	Smoothness and regularity of solutions to PDEs
35A35	Theoretical approximation in context of PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Keywords:

Helmholtz; Laplacian; Neumann screen; Dirichlet screen; boundary integral equations; local finite energy; Wiener-Hopf method; localization; calculus of pseudodifferential operators; existence; uniqueness; regularity; Galerkin schemes; convergence rates; singular elements

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