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Regularization techniques for numerical approximation of PDEs with singularities. (English) Zbl 1035.65085
Summary: The rate of convergence for numerical methods approximating differential equations are often drastically reduced from lack of regularity in the solution. Typical examples are problems with singular source terms or discontinuous material coefficients. We shall discuss the technique of local regularization for handling these problems. New numerical methods are presented and analyzed and numerical examples are given. Some serious deficiencies in existing regularization methods are also pointed out.

MSC:
65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65L20 Stability and convergence of numerical methods for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
35L45 Initial value problems for first-order hyperbolic systems
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35R05 PDEs with low regular coefficients and/or low regular data
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