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 PDF

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

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.

References

Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1749–1779. MR 1885715, DOI 10.1137/S0036142901384162
Blanca Ayuso and L. Donatella Marini, Discontinuous Galerkin methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1391–1420. MR 2485457, DOI 10.1137/080719583
Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. MR 799685, DOI 10.1007/BF01389710
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
Paul Castillo, Bernardo Cockburn, Ilaria Perugia, and Dominik Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 38 (2000), no. 5, 1676–1706. MR 1813251, DOI 10.1137/S0036142900371003
Yanlai Chen and Bernardo Cockburn, Analysis of variable-degree HDG methods for convection-diffusion equations. Part I: general nonconforming meshes, IMA J. Numer. Anal. 32 (2012), no. 4, 1267–1293. MR 2991828, DOI 10.1093/imanum/drr058
Bernardo Cockburn, Bo Dong, and Johnny Guzmán, A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems, Math. Comp. 77 (2008), no. 264, 1887–1916. MR 2429868, DOI 10.1090/S0025-5718-08-02123-6
Bernardo Cockburn, Bo Dong, Johnny Guzmán, Marco Restelli, and Riccardo Sacco, A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems, SIAM J. Sci. Comput. 31 (2009), no. 5, 3827–3846. MR 2556564, DOI 10.1137/080728810
Bernardo Cockburn and Jayadeep Gopalakrishnan, Error analysis of variable degree mixed methods for elliptic problems via hybridization, Math. Comp. 74 (2005), no. 252, 1653–1677. MR 2164091, DOI 10.1090/S0025-5718-05-01741-2
Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319–1365. MR 2485455, DOI 10.1137/070706616
Bernardo Cockburn, Jayadeep Gopalakrishnan, and Francisco-Javier Sayas, A projection-based error analysis of HDG methods, Math. Comp. 79 (2010), no. 271, 1351–1367. MR 2629996, DOI 10.1090/S0025-5718-10-02334-3
Bernardo Cockburn, Johnny Guzmán, and Haiying Wang, Superconvergent discontinuous Galerkin methods for second-order elliptic problems, Math. Comp. 78 (2009), no. 265, 1–24. MR 2448694, DOI 10.1090/S0025-5718-08-02146-7
Bernardo Cockburn, Guido Kanschat, Ilaria Perugia, and Dominik Schötzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal. 39 (2001), no. 1, 264–285. MR 1860725, DOI 10.1137/S0036142900371544
B. Cockburn and W. Qiu. Commuting diagrams for the TNT elements. Math. Comp. (to appear).
B. Cockburn, W. Qiu and K. Shi. Conditions for superconvergence of HDG methods for second-order elliptic problems. Math. Comp., 81:1355–1368, 2012.
Paul Houston, Christoph Schwab, and Endre Süli, Discontinuous $hp$-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal. 39 (2002), no. 6, 2133–2163. MR 1897953, DOI 10.1137/S0036142900374111
N. C. Nguyen, J. Peraire, and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations, J. Comput. Phys. 228 (2009), no. 9, 3232–3254. MR 2513831, DOI 10.1016/j.jcp.2009.01.030
P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR 0483555