Vol. 13, No. 1, 2018

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ISSN: 2157-5452 (e-only)
ISSN: 1559-3940 (print)
Adaptively weighted least squares finite element methods for partial differential equations with singularities

Brian Hayhurst, Mason Keller, Chris Rai, Xidian Sun and Chad R. Westphal

Vol. 13 (2018), No. 1, 1–25
Abstract

The overall effectiveness of finite element methods may be limited by solutions that lack smoothness on a relatively small subset of the domain. In particular, standard least squares finite element methods applied to problems with singular solutions may exhibit slow convergence or, in some cases, may fail to converge. By enhancing the norm used in the least squares functional with weight functions chosen according to a coarse-scale approximation, it is possible to recover near-optimal convergence rates without relying on exotic finite element spaces or specialized meshing strategies. In this paper we describe an adaptive algorithm where appropriate weight functions are generated from a coarse-scale approximate solution. Several numerical tests, both linear and nonlinear, illustrate the robustness of the adaptively weighted approach compared with the analogous standard L2 least squares finite element approach.

Keywords
adaptive finite element methods, weighted norm minimization, singularities
Mathematical Subject Classification 2010
Primary: 65N30
Secondary: 65N12, 35J20, 76D05
Milestones
Received: 23 November 2015
Revised: 6 July 2017
Accepted: 17 September 2017
Published: 17 February 2018
Authors
Brian Hayhurst
Department of Mathematics and Computer Science
Wabash College
Crawfordsville, IN
United States
Mason Keller
Department of Mathematics and Computer Science
Wabash College
Crawfordsville, IN
United States
Chris Rai
Department of Mathematics and Computer Science
Wabash College
Crawfordsville, IN
United States
Xidian Sun
Department of Mathematics
University of Washington
Seattle, WA
United States
Chad R. Westphal
Department of Mathematics and Computer Science
Wabash College
Crawfordsville, IN
United States