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

Abstract

We derive in this paper guaranteed and fully computable a posteriori error estimates for vertex-centered finite-volume-type discretizations of transient linear 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. This algorithm also leads to efficient calculations 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. A collection of numerical experiments illustrates the efficiency of our estimates and the use of the space–time adaptive algorithm.