A Basic Convergence Result for Conforming Adaptive Finite Elements

Abstract

Abstract. We consider the approximate solution with adaptive finite elements of a class of linear boundary value problems, which includes problems of 'saddle point' type. For the adaptive algorithm we suppose the following framework: refinement relies on unique quasi-regular element subdivisions and generates locally quasi-uniform grids, the finite element spaces are conforming, nested, and satisfy the inf-sup conditions, the error estimator is reliable as well as locally and discretely efficient, and marked elements are subdivided at least once. Under these assumptions, we give a sufficient and essentially necessary condition on marking for the convergence of the finite element solutions to the exact one. This condition is not only satisfied by Dörfler's strategy, but also by the maximum strategy and the equidistribution strategy.

Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology

“…where pw, oq P W 1,r 1 0 pΩq dˆLr 1 pΩq{R. Although for the sake of simplicity we restrict ourselves here to residual-based estimates, we note that in principle other a posteriori techniques, such as hierarchical estimates, flux-equilibration or estimates based on local problems, can be used as well; compare with [MSV08,Sie11]. For n P N and G P G let`U n G , P n G˘P VpGqˆQ 0 pGq be the Galerkin approximation defined in (3.20).…”

“…Corollary 22 states the stability properties of the estimator, which are required in order to apply the convergence theory in [Sie11,MSV08]; compare with [Sie11, (2.10b)], for example. The stability of the estimator is also of importance for the efficiency of the estimator.…”

“…The resulting steady non-Newtonian flow problem is then discretized by a mixed finite element method. Guided by an a posteriori error analysis, we propose a numerical method with competing adaptive strategies for the mesh refinement and the approximation of the implicit constitutive law, and we present a rigorous convergence proof generalizing the ideas in [MSV08] and [Sie11]. More precisely, we show that a subsequence of the adaptively generated sequence of discrete approximations converges, in the weak topology of the ambient function space, to a weak solution of the model when 2d…”

Convergence and quasi-optimality of an adaptive finite element method for controlling L 2 errors

“…The convergence properties of AFEM have become the subject of intense theoretical study only in the past few years, however. We refer to [6,[11][12][13][14] for an overview of progress in basic convergence theory for AFEM for linear elliptic problems. Optimal convergence rates were demonstrated in [4,19].…”

Convergence rates for adaptive finite elements

“…The latter work was generalized in various directions. Lately, convergence of adaptive methods with marking strategies other than Dörfler's, for a large class of linear problems with different a posteriori error estimators, and without requiring the marking due to oscillation or the interior node property, was proved in [Morin Siebert Veeser 2007]. The result only leads to asymptotic convergence without an error reduction in every step, which seems to be essential to prove optimality though (see [Stevenson 2006, Cascón et.…”