Ernst P. Stephan

A direct boundary integral equation method for transmission problems

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 Girding’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 spaces for the corresponding Galerkin approximation using finite elements on the boundary only. 0 1985

Boundary integral equations for screen problems in IR3

Here we present a new solution procedure for Helmholtz and Laplacian Neumann screen or Dirichlet screen problems in IR3 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.

Coupling of finite and boundary element methods for an elastoplastic interface problem

A class of transmission problems is considered in which a nonlinear variational problem in one domain is coupled with a linear elliptic problem in a second domain. A typical example is a problem from three-dimensional elasticity theory where an elastoplastic material is embedded into a linear elastic material. The nonlinear problem is given in variational form with a strictly convex functional. The linear elliptic problem is described by boundary integral equations on the coupling boundary. The typical saddle point structure of such problems is analyzed. Galerkin approximations are studied which consist of a finite element approximation in the first domain coupled with a boundary element method on the coupling boundary. The convergence of the Galerkin approximation is based on the saddle-point structure which is shown to hold for the exact as well as the discretized problems.

A hypersingular boundary integral method for two-dimensional screen and crack problems

We analyze hypersingular integral equations on a curved open smooth arc in R 2 that model either curved cracks in an elastic medium or the scattering of acoustic and elastic waves at a hard screen. By using the Mellin transformation we obtain sharp regularity results for the solution of these equations in Sobolev spaces in the form of singular expansions. In particular we show that the expansions do not contain logarithmic singularities

An augmented galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems

Here we present a new solution procedure for Helm-holtz and Laplacian Dirichlet screen and crack problems in IR2 via boundary integral equations of the first kind having as an unknown the jump of the normal derivative across the screen or a crack curve T. Under the assumption of local finite energy we show the equivalence of the integral equations and the original boundary value problem. Via the method of local Mellin transform in [5]-[lo] 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 behaviour near the screen or crack tips.With our integral equations we set up a Galerkin scheme on T and obtain high quasi-optimal convergence rates by using special singular elements besides regular splines as test and trial functions.

A posteriori error estimates for boundary element methods

This paper deals with a general framework for a posteriori error estimates in boundary element methods which is specified for three examples, namely Symms integral equation, an integral equation with a hypersingular operator, and a boundary integral equation for a transmission problem. Based on these estimates, an analog of Eriksson and Johnsons adaptive finite element method is proposed for the h -version of the Galerkin boundary element method for integral equations of the first kind. The efficiency of the approach is shown by numerical experiments which yield almost optimal convergence rates even in the presence of singularities. The construction of an adaptive mesh refinement procedure is of very high practical importance in the numerical analysis of partial differential equations, and we refer to the pioneering work of Babuska and Miller (3) and Eriksson and Johnson (10, 11). Whereas the main features ofadaptivity for finite element methods now seem to be visible, and the door is open to implementation (15), comparably little is known for boundary element methods for integral equations (see e.g. (1, 13, 18, 19,24)). In this paper a new adaptive «-version of the Galerkin discretization for the boundary element method is presented based on a posteriori error estimates. A general framework for these a posteriori error estimates is derived in §2, and three examples are discussed in §§3-5 involving the Dirichlet problem, the Neumann problem (for a closed and an open surface), and a transmission problem for the Laplacian, leading to integral equations with strongly elliptic pseudodifferential operators. Even for smooth data the lack of regularity of the solution near corners (of a polygonal domain Q) leads to poor solutions of the numerical schemes unless appropriate singular functions are incorporated in the trial space or a suitable mesh refinement is used. In practical problems such information is missing, e.g., when we have singular (or nearly singular) data and the main problem is how to balance a graded mesh refinement towards singularities and a global

An improved boundary element Galerkin method for three-dimensional crack problems

In this paper we analyze the solution of crack problems in three-dimensional linear elasticity by equivalent integral equations of the first kind on the crack surface. Besides existence and uniqueness we give sharp regularity results for the solution of these pseudodifferential equations. Two versions of Eskins Wiener-Hopf technique are presented: the first one requires the factorization of matrix-valued symbols which is avoided in the second case. Based on these regularity results we show how to improve the boundary element Galerkin method for our integral equations by using special singular trial functions. We apply the approximation property and inverse assumption of these elements together with duality arguments and derive quasi-optimal asymptotic error estimates in a scale of Sobolev spaces.

Adaptive Boundary Element Methods for Some First Kind Integral Equations

In this paper we present an adaptive boundary element method for the boundary integral equations of the first kind concerning the Dirichlet problem and the Neumann problem for the Laplacian in a two-dimensional Lipschitz domain. For the h-version of the finite element Galerkin discretization of the single layer potential and the hypersingular operator, we derive a posteriori error estimates which guarantee a given bound for the error in the energy norm (up to a multiplicative constant). Following Eriksson and Johnson this yields adaptive algorithms steering the mesh refinement. Numerical examples confirm that our adaptive algorithms yield automatically the expected convergence rate.

Additive schwarz methods for the H-version boundary element method

We study additive Schwarz methods (two level and multilevel) for the h-version boundary element method. Both weakly singular and hypersingular integral equations of the first kind are considered. We prove that the condition numbers of the additive Schwarz operators are bounded independently of the number of levels and number of mesh points. Thus we show that the additive Schwarz method as a parallel preconditioner, which was originally designed for finite element discretisation of differential equations, is also an efficient solver for boundary integral operators, which are non-local operators.

Solution procedures for three-dimensional eddy current problems

Abstract The problem under consideration is that of the scattering of time periodic electromagnetic fields by metallic obstacles. A common approximation here is that in which the metal is assumed to have infinite conductivity. The resulting problem, called the perfect conductor problem, involves solving Maxwells equations in the region exterior to the obstacle with the tangential component of the electric field zero on the obstacle surface. In the interface problem different sets of Maxwell equations must be solved in the obstacle and outside while the tangential components of both electric and magnetic fields are continuous across the obstacle surface. Solution procedures for this problem are given. There is an exact integral equation procedure for the interface problem and an asymptotic procedure for large conductivity. Both are based on a new integral equation procedure for the perfect conductor problem. The asymptotic procedure gives an approximate solution by solving a sequence of problems analogous to the one for perfect conductors.