On the finite element method for elliptic problems with degenerate and singular coefficients

Daniel Arroyo, Alexei Bespalov, Norbert Heuer

    Research output: Contribution to journalArticlepeer-review

    Abstract

    We consider Dirichlet boundary value problems for second order elliptic equations over polygonal domains. The coefficients of the equations under consideration degenerate at an inner point of the domain, or behave singularly in the neighborhood of that point. This behavior may cause singularities in the solution. The solvability of the problems is proved in weighted Sobolev spaces, and their approximation by finite elements is studied. This study includes regularity results, graded meshes, and inverse estimates. Applications of the theory to some problems appearing in quantum mechanics are given. Numerical results are provided which illustrate the theory and confirm the predicted rates of convergence of the finite element approximations for quasi-uniform meshes. © 2006 American Mathematical Society.
    Original languageEnglish
    Pages (from-to)509-537
    Number of pages28
    JournalMathematics of Computation
    Volume76
    Issue number258
    DOIs
    Publication statusPublished - Apr 2007

    Keywords

    • Coulomb field
    • Finite element method
    • Problems with singularities

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