A simple a posteriori estimate on general polytopal meshes with applications to complex porous media flows

Abstract

This paper develops an a posteriori error estimate for lowest-order locally conservative methods on meshes consisting of general polytopal elements. We focus on the ease of implementation of the methodology based on H1-conforming potential reconstructions and H(div,Ω)-conforming flux reconstructions. In particular, the evaluation of our estimates for steady linear diffusion equations merely consists in some local matrix–vector multiplications, where, on each mesh element, the matrices are either directly inherited from the given numerical method, or easily constructed from the element geometry, while the vectors are the flux and potential values on the given element. We next extend our approach to steady nonlinear problems. We obtain a guaranteed upper bound on the total error in the fluxes that is still obtained by local matrix–vector multiplications, with the same element matrices as above. Moreover, the estimate holds true on any linearization and algebraic solver step and allows to distinguish the different error components. Finally, we apply this methodology to unsteady nonlinear coupled degenerate problems describing complex multiphase flows in porous media. Also here, on each step of the time-marching scheme, linearization procedure, and linear algebraic solver, the estimate takes the simple matrix–vector multiplication form and distinguishes the different error components. It leads to an easy-to-implement and fast-to-run adaptive algorithm with guaranteed overall precision, adaptive stopping criteria, and adaptive space and time mesh refinements. Numerous numerical experiments on practical problems in two and three space dimensions illustrate the performance of our methodology.