On the finite element method for elliptic problems with degenerate and singular coefficients |
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Authors: | Daniel Arroyo Alexei Bespalov Norbert Heuer. |
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Affiliation: | Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile ; Computational Center, Far-Eastern Branch of the Russian Academy of Sciences, Khabarovsk, Russia ; BICOM, Department of Mathematical Sciences, Brunel University, Uxbridge UB8 3PH, United Kingdom |
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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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Keywords: | Finite element method problems with singularities Coulomb field. |
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