Numerical calculation of weakly singular surface integrals

We consider semi-analytical and purely numerical integration methods for weakly singular integrals with point singularities on curved smooth surfaces. The methods can be applied to many practical computations in Geodesy, e.g. terrain corrections, Stokes' and Hotines' integral, surface potentials, and the solution of geodetic boundary value problems using integral equations. Current numerical integration techniques are reviewed. The most important semi-analytical and purely numerical techniques are described. Test calcualtions are done and the techniques are compared as regards accuracy and computational efficiency. Semi-analytical methods, which are based on some regularizing parameter transformations, are superior to purely numerical techniques. The best choice are modified polar coordinates defined in the parameter domain with the singularity as pole. Triangular coordinates show similar performance if carefully tuned. Extrapolation techniques and adaptive subdivision techniques behave poorly as regards accuracy and numerical efficiency. Standard integration techniques, which ignore the singularity, completely fail.

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