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 Symm's 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 Johnson's 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

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