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A direct boundary integral equation method for transmission problems. (English) Zbl 0597.35021

A system of integral equations for the field and its normal derivative on the boundary in acoustic or potential scattering by a penetrable homogeneous object in arbitrary dimensions is presented. The system contains the operators of the single and double layer potentials, of the normal derivative of the single layer, and of the normal derivative of the double layer potential. It defines a strongly elliptic system of pseudodifferential operators. It is shown by the method of Mellin transformation that a corresponding property, namely a Gårding’s inequality in the energy norm, holds also in the case of a polygonal boundary of a plane domain. This yields asymptotic quasioptimal error estimates in Sobolev space for the corresponding Galerkin approximation using finite elements on the boundary only.
Reviewer: R.Kreß

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
49M15 Newton-type methods
65N15 Error bounds for boundary value problems involving PDEs
45A05 Linear integral equations
35A22 Transform methods (e.g., integral transforms) applied to PDEs
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