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Collective marking for adaptive least-squares finite element methods with optimal rates

Author: Carsten Carstensen
Journal: Math. Comp. 89 (2020), 89-103
MSC (2010): Primary 65N30, 65N12, 65N15
Published electronically: September 4, 2019
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Abstract: All previously known optimal adaptive least-squares finite element methods (LSFEMs) combine two marking strategies with a separate $ L^2$ data approximation as a consequence of the natural norms equivalent to the least-squares functional. The algorithm and its analysis in this paper circumvent the natural norms in a div-LSFEM model problem with lowest-order conforming and mixed finite element functions and allow for a simple collective Dörfler marking for the first time. A refined analysis provides discrete reliability and quasi-orthogonality in the weaker norms $ L^2\times H^1$ rather than $ H(\operatorname {div})\times H^1$ and replaces data approximation terms by data oscillations. The optimal convergence rates then follow for the lowest-order version from the axioms of adaptivity for the newest-vertex bisection without restrictions on the initial mesh-size in any space dimension.

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Additional Information

Carsten Carstensen
Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany

Keywords: Adaptivity, finite element method, least-squares, nonconforming, a priori, a posteriori, adaptive mesh-refinement, optimal convergence rates, axioms of adaptivity
Received by editor(s): February 24, 2018
Received by editor(s) in revised form: November 23, 2018
Published electronically: September 4, 2019
Additional Notes: This work was supported in the project Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics (CA 151/22-2) by the DFG via the priority program 1748 Reliable Simulation Techniques in Solid Mechanics, Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis
Article copyright: © Copyright 2019 American Mathematical Society