A Posteriori Error Estimates for Finite Volume Approximations

Abstract

Abstract. We present new a posteriori error estimates for the finite volume approximations of elliptic problems. They are obtained by applying functional a posteriori error estimates to natural extensions of the approximate solution and its flux computed by the finite volume method. The estimates give guaranteed upper bounds for the errors in terms of the primal (energy) norm, dual norm (for fluxes), and also in terms of the combined primal-dual norms. It is shown that the estimates provide sharp upper and lower bounds of the error and their practical computation requires solving only finite-dimensional problems.Key words: finite volume methods, elliptic problems, a posteriori error estimates of the functional type

A posteriori error estimates for hierarchical mixed-dimensional elliptic equations

“…where F K,e is the approximated flux trough the edge e of the element K, and E K is the set of edges of the element K. Note that F K,e represents both standard fluxes and, for faces on the boundary to lower-dimensional objects, projections of mortar fluxes. Now the fluxes F K,e can be extended using RTN 0 on each element for each subdomain [18,50]. Finally, we can construct u h by collecting all the tangential and normal fluxes, i.e., u h = [u i,h , λ ij,h ].…”

A posteriori error estimates for mixed finite volume solution of elliptic boundary value problems

“…It is worth remarking that this linear system has been studied in [9] for which the matrix is proven to be symmetric and positive definite. Thus, solving the linear system 7, the computation of (u h , F ) is performed using the relations (6) and 5, whereas v h is calculated using the relation (3b).…”

“…In these works, the authors present a general framework for a posteriori error analysis for finite volume methods applied to convection-diffusion-reaction problems. A posteriori error estimates for finite volume methods have also derived in [16,6,13,5] for the standard two-point finite volume schemes, in [17,2] for DDFV schemes, and in [7] for mimetic finite difference methods. In most of the works mentioned above, the derivation of a posteriori error estimators needs to construct a novel solution which is more regular than the one produced by the scheme.…”

On polynomial robustness of flux reconstructions

“…The extension of these ideas to nonconforming discretizations can be found in e.g. [8], [20]. The quality of the resulting error estimate depends heavily on the choice of the flux reconstruction.…”