Finite elements for elliptic problems with wild coefficients
Computational science, mathematics and software
Finite element methods for semilinear elliptic and parabolic interface problems
Applied Numerical Mathematics
Finite element methods with numerical quadrature for elliptic problems with smooth interfaces
Journal of Computational and Applied Mathematics
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The purpose of this paper is to analyze the error of the finite element method applied to the pressure equation arising in reservoir simulation. We study self-adjoint second-order elliptic equations with discontinuous coefficients and arbitrarily small (but uniformly positive) ellipticity. Under proper conditions on the permeability functions and the source term, we prove error estimates that are independent of the lower bound $\delta$ of the materiel coefficients. These results are based on an extensive regularity analysis of the interface problems of concern. More precisely, we show that the solution of our model problem is piecewise smooth and that the associated Sobolev norms are bounded independently of $\delta$. Finally, the error analysis is illustrated by numerical experiments.