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A posteriori error estimates for boundary element methods. (English) Zbl 0831.65120
The paper presents a new adaptive $$h$$-version of the Galerkin discretization for the boundary element method which is based on a posteriori error estimates. After giving a natural framework for these a posteriori error estimates, three examples are discussed involving the Dirichlet problem, the Neumann problem (for a closed and an open surface), and a transmission problem for the Laplacian.
The approach leads to an upper bound of the global error in energy norms consisting of terms which can be evaluated locally and needs no restriction at all on the mesh for two-dimensional problems. The efficiency of the method is shown by numerical examples which yield almost optimal convergence rates even in the presence of singularities.

##### MSC:
 65N38 Boundary element methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65R20 Numerical methods for integral equations 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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