Functional A Posteriori Error Estimation for Stationary Reaction-Convection-Diffusion Problems

Abstract

-A functional type a posteriori error estimator for the finite element discretization of the stationary reaction-convection-diffusion equation is derived. In case of dominant convection, the solution for this class of problems typically exhibits boundary layers and shock-front like areas with steep gradients. This renders the accurate numerical solution very demanding and appropriate techniques for the adaptive resolution of regions with large approximation errors are crucial. Functional error estimators as derived here contain no mesh-dependent constants and provide guaranteed error bounds for any conforming approximation. To evaluate the error estimator, a minimization problem is solved which does not require any Galerkin orthogonality or any specific properties of the employed approximation space. Based on a set of numerical examples, we assess the performance of the new estimator. It is observed that it exhibits a good efficiency also with convection-dominated problem settings.2010 Mathematical subject classification: 65N30, 65N15, 65J15, 65N22, 65J10.

Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction-diffusion problems

“…As a side result, we also prove in Proposition 5.1 that the weights w K in (1.11) are necessary for robustness of any equilibrated flux estimate involving the terms ε∇u T + ε −1 σ T K whenever σ T is a piecewise polynomial on T (and thus its construction does not involve any submesh), regardless of the precise details of the construction of σ T . This proves that several flux equilibrations proposed in the past cannot be robust with respect to reaction dominance in general (although in many constellations, no loss of robustness may be numerically observed), including those of Repin and Sauter [29], Ainsworth et al [1], Eigel and Samrowski [16], Eigel and Merdon [15], and Vejchodský [32,34,33].…”

“…It is then seen that the inclusion of the weight term w K in Theorem 3.1 is necessary when considering flux equilibrations from vector-valued piecewise polynomial subspaces of H(div, Ω) on the mesh T , regardless of the precise details of the construction of the flux. Examples of flux equilibrations proposed in the past that cannot be robust in general include Repin and Sauter [29], Ainsworth et al [1], Eigel and Samrowski [16], Eigel and Merdon [15], and Vejchodský [32,33,34]. We now present an example of a situation where (5.3) holds and where κh/ε can be arbitrarily large.…”

Functional A Posteriori Error Control for Conforming Mixed Approximations of Coercive Problems with Lower Order Terms

“…Most work on functional type a posteriori error control for the reaction-convection-diffusion problem has concentrated on error estimation for approximations of the primal variable. The error estimates resulting from this research can be found in [4,5,8,9], and some of these results are also exposed in the book [10]. In [8] the authors also derive two-sided estimates for mixed approximations in combined norms.…”

Robust a posteriori error estimates for conforming and nonconforming finite element methods for convection–diffusion problems

“…It is worth mentioning the latest work [32] of Tobiska and Verfürth, where the robust error estimation is extended to other stabilized finite element methods. There are also other proposals in the literature for estimating the error with respect to a priori given (sometimes mesh-dependent) norm, such as residual error indicators [3,23], hierarchical estimates [1], averaging techniques [10] or functional (constant-free) error estimates [13]. All the works mentioned above are only limited to the conforming methods where the discrete solution space is in the primary space.…”