Averaging Techniques for the A Posteriori BEM Error Control for a Hypersingular Integral Equation in Two Dimensions

Averaging techniques or gradient recovery techniques are frequently employed tools for the a posteriori finite element error analysis. Their very recent mathematical justification for partial differential equations allows unstructured meshes and nonsmooth exact solutions. This paper establishes an averaging technique for the hypersingular integral equation on a one-dimensional boundary and presents numerical examples that show averaging techniques can be employed for an effective mesh-refining algorithm. For the discussed test examples, the provided estimator estimates the (in general unknown) error very accurately in the sense that the quotient error/estimator stays bounded with a value close to $1$.

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