Hella Rabus

Optimal adaptive nonconforming FEM for the Stokes problem

This paper presents an optimal nonconforming adaptive finite element algorithm and proves its quasi-optimal complexity for the Stokes equations with respect to natural approximation classes. The proof does not explicitly involve the pressure variable and follows from a novel discrete Helmholtz decomposition of deviatoric functions.

A Natural Adaptive Nonconforming FEM Of Quasi-Optimal Complexity

Abstract In recent years, the question on the convergence and optimality in the context of adaptive finite element methods has been the subject of intensive studies. However, for nonstandard FEMs such as mixed or nonconforming ones, the lack of Galerkins orthogonality requires new mathematical arguments. The presented adap- tive algorithm for the Crouzeix-Raviart finite element method and the Poisson model problem is of quasi-optimal complexity. Furthermore it is natural in the sense that collective marking rather than a separate marking is applied or the estimated error and the volume term.

The adaptive nonconforming FEM for the pure displacement Problem in linear Elasticity is optimal and robust

This paper presents a natural nonconforming adaptive finite element algorithm and proves its quasi-optimal complexity for the pure displacement Navier–Lame equations. The convergence rates are robust with respect to the Lame parameter

On the quasi-optimal convergence of adaptive nonconforming finite element methods in three examples

\lambda \to \infty

Axioms of Adaptivity with Separate Marking for Data Resolution

in the sense that all constants in the quasi-optimal convergence rate stay bounded for almost incompressible materials and so the Stokes equations are covered by our analysis in the limit

Quasi-optimal convergence of AFEM based on separate marking, Part II

\lambda = \infty

An optimal adaptive mixed finite element method

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Axioms of adaptivity for separate marking

Various applications in computational fluid dynamics and solid mechanics motivate the development of reliable and efficient adaptive algorithms for nonstandard finite element methods (FEMs), such as mixed and nonconforming ones. Standard adaptive finite element algorithms consist of the iterative loop of the basic steps Solve, Estimate, Mark and Refine. To reduce the number of degrees of freedom, in adaptive algorithms not all finite element domains are refined successively, which differs from uniform refinement. Instead a selection of finite element domains is marked for refinement on each level based on some refinement indicator. Since some element domains in an automatic mesh refinement may stay relatively coarse, even the analysis of convergence and more importantly the analysis of optimality require new arguments beyond an a priori error analysis. In adaptive algorithms, based on collective marking, a (total) error estimator is used as refinement indicator. For separate marking strategies, this standard scheme may be universalised. The (total) error estimator is split into a volume term and an error estimator term, which estimates the error, but possibly disregards the volume term. Since the volume term is independent of the discrete solution, if there is a poor data approximation the improvement may be realised by a possibly high degree of local mesh refinement. Otherwise, a standard level-oriented mesh refinement based on an error estimator term is performed. This observation results in a natural adaptive algorithm based on separate marking, which is analysed in this thesis. The results of the numerical experiments displayed in this thesis provide strong evidence for the quasi-optimality of the presented adaptive algorithm based on separate marking and for all three model problems. Furthermore its flexibility (in particular the free steering parameter for data approximation) allows a sufficient data approximation in just a few number of levels of the adaptive scheme and at the same time a fast, but optimal, increase of the number of degrees of freedom. This thesis adapts standard arguments for optimal convergence to adaptive algorithms based on separate marking with a possibly high degree of local mesh refinement , and proves quasi-optimality following a general methodology for three model problems, i.e., the Poisson model problem, the pure displacement problem in linear elasticity and the Stokes equations. The numerical experiments confirm the optimal convergence rates.

Mixed Finite Element Method for a Degenerate Convex Variational Problem from Topology Optimization

Mixed finite element methods with flux errors in

Towards adaptive discontinuous Petrov-Galerkin methods

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