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An analysis of the practical DPG method


Authors: J. Gopalakrishnan and W. Qiu
Journal: Math. Comp. 83 (2014), 537-552
MSC (2010): Primary 65N30, 65L12
DOI: https://doi.org/10.1090/S0025-5718-2013-02721-4
Published electronically: May 31, 2013
MathSciNet review: 3143683
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Abstract: We give a complete error analysis of the Discontinuous Petrov Galerkin (DPG) method, accounting for all the approximations made in its practical implementation. Specifically, we consider the DPG method that uses a trial space consisting of polynomials of degree $ p$ on each mesh element. Earlier works showed that there is a ``trial-to-test'' operator $ T$, which when applied to the trial space, defines a test space that guarantees stability. In DPG formulations, this operator $ T$ is local: it can be applied element-by-element. However, an infinite dimensional problem on each mesh element needed to be solved to apply $ T$. In practical computations, $ T$ is approximated using polynomials of some degree $ r > p$ on each mesh element. We show that this approximation maintains optimal convergence rates, provided that $ r\ge p+N$, where $ N$ is the space dimension (two or more), for the Laplace equation. We also prove a similar result for the DPG method for linear elasticity. Remarks on the conditioning of the stiffness matrix in DPG methods are also included.


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Additional Information

J. Gopalakrishnan
Affiliation: Department of Mathematics, Portland State University, P.O. Box 751, Portland, Oregon 97207-0751
Email: gjay@pdx.edu

W. Qiu
Affiliation: The Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455
Address at time of publication: Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong
Email: weifeqiu@cityu.edu.hk

DOI: https://doi.org/10.1090/S0025-5718-2013-02721-4
Keywords: Discontinuous Galerkin, Petrov-Galerkin, DPG method, ultraweak formulation
Received by editor(s): July 21, 2011
Received by editor(s) in revised form: May 23, 2012
Published electronically: May 31, 2013
Additional Notes: Corresponding author: Weifeng Qiu
This work was partly supported by the NSF under grants DMS-1211635 and DMS-1014817. The authors gratefully acknowledge the collaboration opportunities provided by the IMA (Minneapolis) during their 2010-11 program
Article copyright: © Copyright 2013 American Mathematical Society