Convergence of natural adaptive least squares finite element methods

Carsten Carstensen; Eun Jae Park; Philipp Bringmann

doi:10.1007/s00211-017-0866-x

Convergence of natural adaptive least squares finite element methods

Carsten Carstensen, Eun Jae Park, Philipp Bringmann

Department of Computational Science and Engineering

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

Abstract

The first-order div least squares finite element methods provide inherent a posteriori error estimator by the elementwise evaluation of the functional. In this paper we prove Q-linear convergence of the associated adaptive mesh-refining strategy for a sufficiently fine initial mesh with some sufficiently large bulk parameter for piecewise constant right-hand sides in a Poisson model problem. The proof relies on some modification of known supercloseness results to non-convex polygonal domains plus the flux representation formula. The analysis is carried out for the lowest-order case in two-dimensions for the simplicity of the presentation.

Original language	English
Pages (from-to)	1097-1115
Number of pages	19
Journal	Numerische Mathematik
Volume	136
Issue number	4
DOIs	https://doi.org/10.1007/s00211-017-0866-x
Publication status	Published - 2017 Aug 1

Bibliographical note

Funding Information:
This research was supported by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 “Reliable simulation techniques in solid mechanics. Development of non- standard discretization methods, mechanical and mathematical analysis” under the project “Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics” (CA 151/22-1). This research was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology NRF 2011-0030934 and NRF-2015R1A5A1009350.

Publisher Copyright:
© 2017, Springer-Verlag Berlin Heidelberg.

All Science Journal Classification (ASJC) codes

Computational Mathematics
Applied Mathematics

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10.1007/s00211-017-0866-x

Cite this

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abstract = "The first-order div least squares finite element methods provide inherent a posteriori error estimator by the elementwise evaluation of the functional. In this paper we prove Q-linear convergence of the associated adaptive mesh-refining strategy for a sufficiently fine initial mesh with some sufficiently large bulk parameter for piecewise constant right-hand sides in a Poisson model problem. The proof relies on some modification of known supercloseness results to non-convex polygonal domains plus the flux representation formula. The analysis is carried out for the lowest-order case in two-dimensions for the simplicity of the presentation.",

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AB - The first-order div least squares finite element methods provide inherent a posteriori error estimator by the elementwise evaluation of the functional. In this paper we prove Q-linear convergence of the associated adaptive mesh-refining strategy for a sufficiently fine initial mesh with some sufficiently large bulk parameter for piecewise constant right-hand sides in a Poisson model problem. The proof relies on some modification of known supercloseness results to non-convex polygonal domains plus the flux representation formula. The analysis is carried out for the lowest-order case in two-dimensions for the simplicity of the presentation.

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Convergence of natural adaptive least squares finite element methods

Abstract

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