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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Analysis of variable-degree HDG methods for Convection-Diffusion equations. Part II: Semimatching nonconforming meshes


Authors: Yanlai Chen and Bernardo Cockburn
Journal: Math. Comp. 83 (2014), 87-111
MSC (2010): Primary 65M60, 65N30
Published electronically: May 16, 2013
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Abstract: In this paper, we provide a projection-based analysis of the $ h$-version of the hybridizable discontinuous Galerkin methods for convection-diffusion equations on semimatching nonconforming meshes made of simplexes; the degrees of the piecewise polynomials are allowed to vary from element to element. We show that, for approximations of degree $ k$ on all elements, the order of convergence of the error in the diffusive flux is $ k+1$ and that of a projection of the error in the scalar unknown is $ 1$ for $ k=0$ and $ k+2$ for $ k>0$. We also show that, for the variable-degree case, the projection of the error in the scalar variable is $ h$ times the projection of the error in the vector variable, provided a simple condition is satisfied for the choice of the degree of the approximation on the elements with hanging nodes. These results hold for any (bounded) irregularity index of the nonconformity of the mesh. Moreover, our analysis can be extended to hypercubes.


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Additional Information

Yanlai Chen
Affiliation: Department of Mathematics, University of Massachusetts Dartmouth, 285 Old Westport Road, North Dartmouth, Massachusetts 02747
Email: yanlai.chen@umassd.edu

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: cockburn@math.umn.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2013-02711-1
PII: S 0025-5718(2013)02711-1
Received by editor(s): July 5, 2011
Received by editor(s) in revised form: December 21, 2011, and May 4, 2012
Published electronically: May 16, 2013
Additional Notes: The research of the second author was partially supported by the National Science Foundation (grant DMS-0712955).
Article copyright: © Copyright 2013 American Mathematical Society