M. Aurada, M. Feischl, T. Führer, M. Karkulik, M. Page,D. Praetorius:

"What is quasi-optimality of adaptive FEM?";

Keynote Lecture: TCSE Vienna Workshop 2012, Trends in Computational Science and Engineering, Wien (invited); 02-13-2012 - 02-14-2012.

Usually the accuracy of numerical results suffers from singularities

of the given data and/or the (unknown) exact solution. One remedy is

to appropriately grade the underlying mesh towards these singularities. In

many codes, this mesh adaptation is done automatically by use of certain

(heuristic) refinement indicators. Empirically, most of these

adaptive strategies perform very well in the sense that the slope

of number of unknowns versus corresponding error shows the

best possible decay. Such an observation is of practical relevance

since each simulation aims for the best possible solution with respect

to limited resources like computational time and/or memory requirements.

In recent years and for a certain class of elliptic differential equations,

there has been a major breakthrough in the mathematical understanding of

this empirical observation. In our talk, we aim to state "optimal convergence

behaviour" of the numerical scheme in a mathematical sense. We discuss

the progress made in the recent years and highlight some of the mathematical

ingredients for such an optimality proof.

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