Adaptivite Least Squares Finite Element Methods
Speaker
Prof. Carsten Carstensen
Humboldt-Universität zu Berlin, Germany
Abstract

The leasts squares finite element schemes are popular because they lead to symmetric and positive definite discrete problems with seemingly no stability conditions. Although their similarity with mixed finite element method has been observed in [1], the natural choice of the localized least squares residual as an mesh-refining criterion is not justified. It has been observed that the associated adaptive algorithm it is convergence for a bulk parameter very large close to one [2]. This is in contradiction to the methodology of optimal rates where this parameter has to be sufficiently small [3,4]. One remedy in the analysis of mixed finite element schemes, of leasts squares finite element schemes and for the discontinuous Petrov-Galerkin methodology is the usage of separate marking and alternative a~posteriori error estimates as in [5]. Mixed finite element methods with flux errors in $H(\ddiv)$-norms and div-least-squares finite element methods require the separate marking strategy in obligatory adaptive mesh-refining. The refinement indicator $\sigma_\ell^2(K)=\eta_\ell^2(K)+\mu^2(K)$ of a finite element domain $K$ in a triangulation $\Tl \ell$ on the level $\ell$ consists of some residual-based error estimator $\eta_\ell$ with some reduction property under local mesh-refining and some data approximation error $\mu_\ell$. Separate marking (\textsc{Safem}) means either D\"orfler marking (or bulk chasing) if $\mu_\ell^2 \leq \kappa \eta_\ell^2$ and otherwise an optimal data approximation algorithm run with controlled accuracy [4,5]. The presentation explains the setting and results of more practical applications for the Stokes problem [6,7] and linear elasticity [8] and ends with an outlook on the possibilities for nonlinear problems.

References:
[1] J. Brandts, Y. Chen, and J. Yang, A note on least-squares mixed finite elements in relation to standard and mixed finite elements, IMA J. Numer. Anal., 26 (2006), pp. 779-789
[2] P. Bringmann, C. Carstensen, and E. J. Park. Convergence of natural adaptive least-squares finite element methods. Numer. Math. (2017)
[3] C.~Carstensen, M.~Feischl, M.~Page, and D.~Praetorius. Axioms of adaptivity. Comput.\ Methods Appl.\ Math., 67(6):1195--1253, 2014.
[4] C.~Carstensen and H.~Rabus. Axioms of adaptivity for separate marking. arXiv:1606.02165v1 SINUM accepted for publication 2017.
[5] C.~Carstensen and E.-J. Park. Convergence and optimality of adaptive least squares finite element methods. SIAM J. Numer. Anal., 53:43--62, 2015.
[6] P. Bringmann and C. Carstensen. An adaptive least-squares FEM for the Stokes equations with optimal convergence rates. Numer. Math., 135(2):459-492, 2017.
[7] P. Bringmann and C. Carstensen. h-adaptive least-squares finite element method for the 2d stokes equations of any order with optimal convergence rates. Computers and Mathematics with Applications, 2017