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The Finite Element Method of High Degree of Accuracy for Boundary Value Problem with Singularity

Author

Listed:
  • Viktor A. Rukavishnikov

    (Computing Center of Far Eastern Branch Russian Academy of Sciences, Kim Yu Chen Str. 65, 680000 Khabarovsk, Russia
    These authors contributed equally to this work.)

  • Elena I. Rukavishnikova

    (Computing Center of Far Eastern Branch Russian Academy of Sciences, Kim Yu Chen Str. 65, 680000 Khabarovsk, Russia
    These authors contributed equally to this work.)

Abstract

Mathematical models of fracture physics and mechanics are boundary value problems for differential equations and systems of equations with a singularity. There are two classes of problems with a singularity: with coordinated and uncoordinated degeneracy of the input data, depending on the behavior of the coefficients of the equation. Finite element methods with the first order of convergence rate O ( h ) have been created to find an approximate solution to these problems. We construct a scheme of the weighted finite element method of high degree of accuracy for the boundary value problem with uncoordinated degeneracy of the input data and singularity of the solution. The rate of convergence of an approximate solution of the proposed finite element method to the exact R ν -generalized solution in the weight set W 2 , ν + β 2 + 2 1 ( Ω , δ ) is investigated. The estimation of finite element approximation O ( h 2 ) is established.

Suggested Citation

  • Viktor A. Rukavishnikov & Elena I. Rukavishnikova, 2023. "The Finite Element Method of High Degree of Accuracy for Boundary Value Problem with Singularity," Mathematics, MDPI, vol. 11(15), pages 1-11, July.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:15:p:3272-:d:1202236
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