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A projection-based error analysis of HDG methods for Timoshenko beams

Authors: Fatih Celiker, Bernardo Cockburn and Ke Shi
Journal: Math. Comp. 81 (2012), 131-151
MSC (2010): Primary 65M60, 65N30, 35J30
Published electronically: July 11, 2011
MathSciNet review: 2833490
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Abstract: In this paper, we give the first a priori error analysis of the hybridizable discontinuous Galerkin (HDG) methods for Timoshenko beams. The analysis is based on the use of a projection especially designed to fit the structure of the numerical traces of the HDG method. This property allows us to prove in a very concise manner that the projection of the errors is bounded in terms of the distance between the exact solution and its projection. The study of the influence of the stabilization function on the approximation is then reduced to the study of how they affect the approximation properties of the projection in a single element. Surprisingly, and unlike any other discontinuous Galerkin method, this can be done without assuming any positivity property of the stabilization function of the HDG method. We apply this approach to HDG methods using polynomials of degree $k\ge 0$ in all of the unknowns, and show that the projection of the error in each of them superconverges with order $k+2$ when $k \ge 1$ and converges with order $1$ for $k=0$. As a result, we show that the HDG methods converge with optimal order $k+1$ for all the unknowns, and that they are free from shear locking. Finally, we show that all of the numerical traces converge with order $2k+1$. Numerical experiments validating these results are shown.

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Additional Information

Fatih Celiker
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, Minnesota 55455

Ke Shi
Affiliation: School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, Minnesota 55455
MR Author ID: 904733

Keywords: Discontinuous Galerkin methods, Timoshenko beams, hybridization
Received by editor(s): April 11, 2010
Received by editor(s) in revised form: October 12, 2010
Published electronically: July 11, 2011
Additional Notes: The second author was partially supported by the National Science Foundation (Grant DMS-0712955) and by the Minnesota Supercomputing Institute. Part of this work was done when this author was visiting the Research Institute for Mathematical Sciences, Kyoto University, Japan, during the Fall of 2009.
Article copyright: © Copyright 2011 American Mathematical Society