European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:30   MS3: Adaptivity, Regularity and Fast Approximation (Part 1)
Chair: Markus Weimar
25 mins
Near-Best Adaptive Approximations
Peter Binev
Abstract: The talk features some recent theoretical results about near-best adaptive approximations and their relations to obtaining optimal convergence rates for different types of adaptive finite element methods.
25 mins
Tree approximation with conforming meshes
Peter Binev, Francesca Fierro, Andreas Veeser
Abstract: Tree approximation with conforming meshes Consider approximants over conforming (or face-to-face) meshes that are generated by means of bisection of simplices. Then the tree approximation algorithms by Binev and DeVore (2004) and Binev (2018) construct near best approximants with quasi-optimal complexity. Doing so, the decisions in the algorithm do not take into account the requirement of conformity. This entails that the near best constant depends on the completion process which turns a nonconforming mesh into a conforming one. We shall generalize the tree approximation algorithm form Binev (2018) to take into account the conformity of the generated meshes, thereby avoiding the aforementioned dependence.
25 mins
A priori regularity results for discrete solutions to elliptic equations on graded meshes
Toni Scharle, Lars Diening, Endre Süli
Abstract: We will develop discrete versions of De-Giorgi-type regularity results for piecewise linear discrete solutions to elliptic equations on graded meshes. To achieve this, we have to find Caccioppoli-type inequalities for truncated solutions. This will lead to quantitative Hölder-estimates in the case of uniformly elliptic linear equations and local point-wise bounds for p-Laplacian systems.
25 mins
Optimal adaptivity for the Stokes problem
Michael Feischl
Abstract: We prove that the a standard adaptive algorithm for the Taylor-Hood discretization of the stationary Stokes problem converges with optimal rate. This is done by developing an abstract framework for indefinite problems which allows us to prove quasi-orthogonality for indefinite and non-symmetric problems. This property is the main obstacle towards the optimality proof and therefore is the main focus of this work. The key ingredient is a new connection between the mentioned quasi-orthogonality and the LU-factorization of infinite matrices.