All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable

摘要

All first-order averaging or gradient-recovery operators for lowest-order finite element methods are shown to allow for an efficient a posteriori error estimation in an isotropic, elliptic model problem in a bounded Lipschitz domain Omega in R-d. Given a piecewise constant discrete flux p(h) is an element of P-h (that is the gradient of a discrete displacement) as an approximation to the unknown exact flux p (that is the gradient of the exact displacement), recent results verify efficiency and reliability ofetaM :=min{Ph-qh(L2(Omega)) : qh is an element of Qg}in the sense that eta(M) is a lower and upper bound of the flux error p-p(h)(L2 (Omega)) up to multiplicative constants and higher-order terms. The averaging space Q(h) consists of piecewise polynomial and globally continuous finite element functions in d components with carefully designed boundary conditions. The minimal value eta(M) is frequently replaced by some averaging operator A : P-h --> Q(h) applied within a simple post-processing to p(h). The result (qh) := A(ph) is an element of Q(h) provides a reliable error bound with eta(M) less than or equal to eta(A) := (ph)-A(ph)(L2(Omega)).This paper establishes eta(A) less than or equal to C-eff eta(M) and so equivalence of eta(M) and eta(A). This implies efficiency of eta(A) for a large class of patchwise averaging techniques which includes the ZZ-gradient-recovery technique. The bound C-eff less than or equal to 3: 88 established for tetrahedral P-1 finite elements appears striking in that the shape of the elements does not enter: The equivalence eta(A) approximate to eta(M) is robust with respect to anisotropic meshes. The main arguments in the proof are Ascoli's lemma, a strengthened Cauchy inequality, and elementary calculations with mass matrices.