A posteriori error estimates for combined finite volume-finite element discretizations of reactive transport equations on nonmatching grids

摘要

We derive fully computable a posteriori error estimates for vertex-centered finite volume-type discretizations of transient convection-diffusion-reaction equations. Our estimates enable actual control of the error measured either in the energy norm or in the energy norm augmented by a dual norm of the skew-symmetric part of the differential operator. Lower bounds, global-in-space but local-in-time, are also derived. These lower bounds are fully robust with respect to convection or reaction dominance and the final simulation time in the augmented norm setting. On the basis of the derived estimates, we propose an adaptive algorithm which enables to automatically achieve a user-given relative precision. Moreover, this algorithm leads to optimal efficiency as it balances the time and space error contributions. As an example, we apply our estimates to the combined finite volume-finite element scheme, including such features as use of mass lumping for the time evolution or reaction terms, of upwind weighting for the convection term, and discretization on nonmatching meshes possibly containing nonconvex and non-star-shaped elements. Numerical experiments illustrate the theoretical developments.