Christian Kreuzer

Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method

We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As is customary in practice, the AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that the AFEM is a contraction, for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops. This geometric decay is instrumental to derive the optimal cardinality of the AFEM. We show that the AFEM yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.

Linear Convergence of an Adaptive Finite Element Method for the

We study an adaptive finite element method for the

p

p

-Laplacian Equation

-Laplacian like PDEs using piecewise linear, continuous functions. The error is measured by means of the quasi norm of Barrett and Liu. We provide residual based error estimators without a gap between the upper and lower bound. We show linear convergence of the algorithm which is similar to the one of Morin, Nochetto, and Siebert. All results are obtained without extra marking for the oscillation.

Decay rates of adaptive finite elements with Dörfler marking

We investigate the decay rate for an adaptive finite element discretization of a second order linear, symmetric, elliptic PDE. We allow for any kind of estimator that is locally equivalent to the standard residual estimator. This includes in particular hierarchical estimators, estimators based on the solution of local problems, estimators based on local averaging, equilibrated residual estimators, the ZZ-estimator, etc. The adaptive method selects elements for refinement with Dörfler marking and performs a minimal refinement in that no interior node property is needed. Based on the local equivalence to the residual estimator we prove an error reduction property. In combination with minimal Dörfler marking this yields an optimal decay rate in terms of degrees of freedom.

Finite Element Approximation of Steady Flows of Incompressible Fluids with Implicit Power-Law-Like Rheology

We develop the analysis of finite element approximations of implicit power-law-like models for viscous incompressible fluids. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multivalued, maximal monotone

Instance Optimality of the Adaptive Maximum Strategy

r

Analysis of an adaptive Uzawa finite element method for the nonlinear Stokes problem

-graph with

A note on why enforcing discrete maximum principles by a simple a posteriori cutoff is a good idea

1<r<\infty

Convergence of adaptive discontinuous galerkin methods

. Using a variety of weak compactness techniques, including Chacons biting lemma and Young measures, we show that a subsequence of the sequence of finite element solutions converges to a weak solution of the problem as the finite element discretization parameter