Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients

Martin Vohralík (1)

1 LJLL - Laboratoire Jacques-Louis Lions

We study in this paper a posteriori error estimates for $H^1$-conforming numerical approximations of diffusion problems with a diffusion coefficient piecewise constant on the mesh cells but arbitrarily discontinuous across the interfaces between the cells. Our estimates give a global upper bound on the error measured either as the energy norm of the difference between the exact and approximate solutions, or as a dual norm of the residual. They are guaranteed, meaning that they feature no undetermined constants. (Local) lower bounds for the error are also derived. Herein, only generic constants independent of the diffusion coefficient appear, whence our estimates are fully robust with respect to the jumps in the diffusion coefficient. In particular, no condition on the diffusion coefficient like its monotonous increasing along paths around mesh vertices is imposed, whence the present results also include the cases with singular solutions. For the energy error setting, 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 scheme. 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 affine finite element ones. For the dual norm setting, no such a requirement is necessary. Our estimates are based on $\tH(\dv)$-conforming flux reconstruction obtained thanks to the local conservativity of all the studied methods on the dual grids, which we recall in the paper; mutual relations between the different methods are also recalled. Numerical experiments are presented in confirmation of the guaranteed upper bound, full robustness, and excellent efficiency of the derived estimators.

Mots clés: Finite volume method; Finite element method; Finite difference method; Discontinuous coefficients; Harmonic averaging; A posteriori error estimates; Guaranteed upper bound; Robustness

Domaines: Analyse numérique [math.NA]

Martin Vohralík. Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients. Journal of Scientific Computing, 2011, 46 (3), pp.397-438. ⟨10.1007/s10915-010-9410-1⟩. ⟨hal-00235810v4⟩