Section: New Results

Convergence of adaptive finite element algorithms

Participants : Roland Becker, Shipeng Mao, David Trujillo.

Figure 4. Solution with rough right-hand-side in a corner domain and adaptively refined mesh (from [12] ).

Adaptive finite element methods are becoming a standard tool in numerical simulations, and their application in CFD is one of the main topics of CONCHA, see Section  3.7 . Such methods are based on a posteriori error estimates of the discretization, avoiding dependance on the continuous solution, as known from a priori error estimates. The estimator is used in an adaptive loop by means of a local mesh refinement algorithm. The mathematical theory of these algorithms has for a long time be bounded to the proof of upper and lower bounds, but has made important improvements in recent years. For illustration, a typical sequence of adaptively refined meshes on an L -shaped domain is shown in Figure 4 .

The theoretical analysis of mesh-adaptive methods, even in the most standard case of the Poisson problem, is in its infancy. The first important results in this direction concern simply the convergence of the sequence of solution generated by the algorithm (the standard a priori error analysis does not apply since the global mesh-size does not necessarily go to zero). In order to the so, an unavoidable data-oscillation term has to be treated in addition to the error estimator [60] . These result do not say anything about the convergence speed, that is the number of unknowns required to achieve a given accuracy. Such complexity estimates are the subject of active research, the first fundamental result in this direction is [45] .

Our first contribution [5] to this field has been the introduction of a new adaptive algorithm which makes use of an adaptive marking strategy, which refines according to the data oscillations only if they are by a certain factor larger then the estimator. This algorithm allows us to prove geometric convergence and quasi-optimal complexity, avoiding additional iteration as used before [65] .

We have extended our results to the case of mixed FE [43] , as well as nonconforming FE [36] . In these cases, a major additional difficulty arises from the fact that the orthogonality relation known from continuous FEM does not hold, either due to the saddle-point formulation or due to the non-nested discrete spaces. In addition, we have considered the case of incomplete solution of the discrete systems. To this end, we have developed a simple adaptive stopping criterion based on comparison of the iteration error with the discretization error estimator [12] .

A further generalization has been to AFEM on quadrilateral meshes with local refinement allowing for hanging nodes [44] . Three major difficulties had to be overcome. First, the normal derivative of a bilinear function is not constant on an edge, and this makes the standard lower bound estimate, used for example in [60] , [65] , unavailable. We have replaced this crucial ingredient by an estimate on the decrease of the estimator under mesh refinement. A further technical point is the fact that the laplacian of an iso-parametric Q1 finite element function is not zero in the interior of the elements. Finally, the complexity estimate for the adaptive solution algorithm relies on a complexity estimate for the local refinement algorithm (notice that additional triangles/quadrilaterals have to be refined in order to fulfill certain criteria). Such an estimate seemed so far only available for the so-called 'newest vertex algorithm', which uses iterated bisection. We have obtained a similar estimate for local refinement of quadrilateral meshes with hanging nodes. The refinement algorithm is constrained to fulfill the regularity assumption that the difference in refinement level of quadrilaterals surrounding a given node is not larger then one.

Recently, we have been able to extend the above mentioned result on quasi-optimality to the Stokes equations (under review). These results have been presented in [17] , [29] .

Our theoretical studies, which are motivated by the aim to develop better adaptive algorithms, have been accompanied by software implementation with the Concha library, see Section  5.1 . It hopefully opens the door to further theoretical an experimental studies. We are actually concerned with generalizations to constant-free estimators, hyperbolic equations, and goal-oriented error estimation [2] .

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