A reliable and efficient a posteriori error estimation plays a key role in the adaptive finite or boundary element methods. In this paper some a posteriori error estimates for boundary element methods, based on classical and hypersingular residuals, are discussed. The residuals are defined as the differences between two sides of the boundary integral equations when the exact solution is replaced by the approximate solution. Three kinds of the residuals are compared here. Numerical examples show some advantages of the hypersingular residuals, especially obtained from the natural boundary integral equations. It is a good o posteriori error indicator for some boundary element methods.
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