Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients
Résumé
We study in this paper a posteriori error estimates for H1-conforming numerical approximations of diffusion problems with a scalar, piecewise constant, and arbitrarily discontinuous diffusion coefficient. We derive estimators for the energy norm which give a guaranteed global upper bound in the sense that they feature no undetermined constants. Local lower bounds, up to constants independent of the diffusion coefficient, are also derived. In particular, no condition on the diffusion coefficient like its monotonous increasing along paths around mesh vertices is imposed, whence the present results are fully robust and include also the cases with singular solutions. The key requirement turns out to be that the diffusion coefficient is piecewise constant on dual cells associated with the vertices of an original simplicial mesh and that harmonic averaging is used in the numerical scheme definition. This is the usual case, e.g., for the cell-centered finite volume method, included in our analysis as well as the vertex-centered finite volume, finite difference, and continuous piecewise linear finite element ones. Our estimates consist of two estimators, one comparing a reconstructed H(div)-conforming diffusive flux with the diffusive flux given by the piecewise linear representation of the approximate solution and one representing an improved residual. They are based on the local conservativity of all the studied methods on the dual grids, which we recall in the paper, as well as their mutual relations. Numerical experiments confirm the guaranteed upper bound, full robustness, and excellent efficiency of the presented estimators, which may still be improved by a negligible-cost local minimization.
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