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

Authors: Daniel Arroyo, Alexei Bespalov and Norbert Heuer
Journal: Math. Comp. 76 (2007), 509-537
MSC (2000): Primary 65N30, 65N15
Published electronically: October 17, 2006
MathSciNet review: 2291826
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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.

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Additional Information

Daniel Arroyo
Affiliation: Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Alexei Bespalov
Affiliation: Computational Center, Far-Eastern Branch of the Russian Academy of Sciences, Khabarovsk, Russia

Norbert Heuer
Affiliation: BICOM, Department of Mathematical Sciences, Brunel University, Uxbridge UB8 3PH, United Kingdom

Keywords: Finite element method, problems with singularities, Coulomb field.
Received by editor(s): November 17, 2003
Received by editor(s) in revised form: July 21, 2005
Published electronically: October 17, 2006
Additional Notes: The first author was supported by Fondecyt project 1010220, Chile.
The second author was supported by the Russian Foundation for Basic Research project no. 01–01–00375 and by the FONDAP Program in Applied Mathematics, Chile.
The third author was supported by the FONDAP Program in Applied Mathematics and Fondecyt projects 1010220, 1040615, Chile.
Article copyright: © Copyright 2006 American Mathematical Society