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 H 1-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.

Adaptive solution of linear systems of equations based on a posteriori error estimators

“…We remind that PCG is known to minimize the global energy norm of the error ||x − x (i) || A , which is also equal to the algebraic error on the whole domain Ω, following (5). Yet, as expressed in Corollary 1, a concentrated algebraic error on a subdomain Ω 1 implies that the A L -inner product of the error is dominant, and so will be the L-term, according to (21). This is why they should be reduced for an efficient decrease of the energy norm of the error.…”

“…A common drawback of the above mentioned, rigorously justified estimators is their evaluation cost, which is typically (very) high with respect to the cost of an algebraic solver iteration. However, recent work [21] has resulted in the development of a posteriori estimates that can be easily coded, cheaply evaluated, and efficiently used in practical simulations providing a guaranteed control over different error components. They have confirmed that the computation of error estimators can be accessible even within non-academic contexts.…”

Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach. Recovering mass balance in any situation

“…Finally, Appendix C presents the application of our mass-conservative total flux reconstruction in H(div, Ω) to a challenging two-phase porous media flow problem with a finite volume fully implict/iterative coupling discretization. Applications to other problems, namely when deriving guaranteed upper bounds on the total error in presence of inexact solvers, have already been considered in [34,35] to steady and unsteady variational inequalities, in [26] to eigenvalue problems, in [60] to goal-oriented error estimates, and in [87,2] to degenerate multiphase (multicompositional) flows.…”

Goal-oriented a posteriori error estimation for conforming and nonconforming approximations with inexact solvers

“…The finite volume method we consider is defined on a possibly nonmatching mesh of general polygonal or polyhedral elements, popular in porous media applications. To cast this setting in the framework of the present paper, we follow [48] and in particular suppose that there exists a virtual simplicial submesh of the polytopal mesh which is matching, shape-regular, and such that any polytopal element is covered by a patch of simplices. For a fast evaluation of the estimators η i h andη i h , we proceed as in [48,Theorem 3.12].…”

“…To cast this setting in the framework of the present paper, we follow [48] and in particular suppose that there exists a virtual simplicial submesh of the polytopal mesh which is matching, shape-regular, and such that any polytopal element is covered by a patch of simplices. For a fast evaluation of the estimators η i h andη i h , we proceed as in [48,Theorem 3.12]. Following Theorem 4.13, the error components are then distinguished as…”