Analysis of HDG methods for Stokes flow

Authors:
Bernardo Cockburn, Jayadeep Gopalakrishnan, Ngoc Cuong Nguyen, Jaume Peraire and Francisco-Javier Sayas

Journal:
Math. Comp. **80** (2011), 723-760

MSC (2010):
Primary 65N30, 65M60, 35L65

DOI:
https://doi.org/10.1090/S0025-5718-2010-02410-X

Published electronically:
September 2, 2010

MathSciNet review:
2772094

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we analyze a hybridizable discontinuous Galerkin method for numerically solving the Stokes equations. The method uses polynomials of degree for all the components of the approximate solution of the gradient-velocity-pressure formulation. The novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method. It renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges. As a consequence, we prove that the approximations of the velocity gradient, the velocity and the pressure converge with the optimal order of convergence of in for any . Moreover, taking advantage of the superconvergence properties of the velocity, we introduce a new element-by-element postprocessing to obtain a new velocity approximation which is exactly divergence-free, div-conforming, and converges with order for and with order for . Numerical experiments are presented which validate the theoretical results.

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Additional Information

**Bernardo Cockburn**

Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Email:
cockburn@math.umn.edu

**Jayadeep Gopalakrishnan**

Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611–8105

Email:
jayg@math.ufl.edu

**Ngoc Cuong Nguyen**

Affiliation:
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge Massachusetts 02139

Email:
cuongng@mit.edu

**Jaume Peraire**

Affiliation:
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge Massachusetts 02139

Email:
peraire@mit.edu

**Francisco-Javier Sayas**

Affiliation:
Departamento de Matemática Aplicada, Universidad de Zaragoza, 50009 Zaragoza, Spain

Email:
sayas002@umn.edu

DOI:
https://doi.org/10.1090/S0025-5718-2010-02410-X

Keywords:
Stokes flow,
mixed methods,
discontinuous Galerkin methods,
hybridized methods,
Lagrange multipliers.

Received by editor(s):
July 29, 2009

Received by editor(s) in revised form:
January 5, 2010

Published electronically:
September 2, 2010

Additional Notes:
The first author was supported in part by the National Science Foundation (Grant DMS-0712955) and by the University of Minnesota Supercomputing Institute

The second author was supported in part by the National Science Foundation under grants DMS-0713833 and SCREMS-0619080

The third author was supported in part by the Singapore-MIT Alliance

The fourth author was supported in part by the Singapore-MIT Alliance.

The fifth author was a Visiting Professor of the School of Mathematics, University of Minnesota, during the development of this work. He was partially supported by MEC/FEDER Project MTM2007–63204 and Gobierno de Aragón (Grupo PDIE)

Article copyright:
© Copyright 2010
American Mathematical Society