Numerical approximations of singular source terms in differential equations.(English)Zbl 1115.76392

Summary: Singular terms in differential equations pose severe challenges for numerical approximations on regular grids. Regularization of the singularities is a very useful technique for their representation on the grid. We analyze such techniques for the practically preferred case of narrow support of the regularizations, extending our earlier results for wider support. The analysis also generalizes existing theory for one dimensional problems to multi-dimensions. New high order multidimensional techniques for differential equations and numerical quadrature are introduced based on the analysis and numerical results are presented. We also show that the common use of distance functions in level-set methods to extend one dimensional regularization to higher dimensions may produce $$O(1)$$ errors.

MSC:

 76M25 Other numerical methods (fluid mechanics) (MSC2010) 74S30 Other numerical methods in solid mechanics (MSC2010) 65N99 Numerical methods for partial differential equations, boundary value problems
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