On the finite element method for elliptic problems
with degenerate and singular coefficients
Daniel Arroyo, Alexei Bespalov and Norbert Heuer
Math. Comp. 76 (258), 509-537, 2007.
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. 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.