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

Chen, Yanlai; Cockburn, Bernardo

doi:10.1090/S0025-5718-2013-02711-1

Analysis of variable-degree HDG methods for convection-diffusion equations. Part II: Semimatching nonconforming meshes
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by Yanlai Chen and Bernardo Cockburn;

Math. Comp. 83 (2014), 87-111

DOI: https://doi.org/10.1090/S0025-5718-2013-02711-1

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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