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 | References | Similar Articles | Additional Information

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 on each mesh element. Earlier works showed that there is a ``trial-to-test'' operator , which when applied to the trial space, defines a test space that guarantees stability. In DPG formulations, this operator is local: it can be applied element-by-element. However, an infinite dimensional problem on each mesh element needed to be solved to apply . In practical computations, is approximated using polynomials of some degree on each mesh element. We show that this approximation maintains optimal convergence rates, provided that , where 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