An a posteriori error estimate for the unsymmetric coupling of FEM and BEM

The coupling of the finite element method (FEM) and boundary element method (BEM) can be performed in at least two different ways leading to a symmetric or an unsymmetric discrete problem. While the symmetric coupling is proved to converge in a large number of situations, this is known for the unsymmetric coupling only in case of a smooth boundary or under some improper restrictions on the meshsizes. Since the unsymmetric coupling is less expensive, it is used by engineers and seemingly gives satisfactory numerical results. Besides the lack of a priori error control one might ask for the justification of the actual numerical output after its computation. In this note we firstly present such a tool and prove a posteriori error estimates for the unsymmetric coupling. Since the computable upper error bound is local, we may use this information to steer the automatic mesh-refinement. The resulting adaptive algorithm is illustrated in a numerical example.

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