A projection-based error analysis of HDG methods

Cockburn, Bernardo; Gopalakrishnan, Jayadeep; Sayas, Francisco-Javier

doi:10.1090/S0025-5718-10-02334-3

A projection-based error analysis of HDG methods

Authors: Bernardo Cockburn, Jayadeep Gopalakrishnan and Francisco-Javier Sayas
Journal: Math. Comp. 79 (2010), 1351-1367
MSC (2010): Primary 65M60, 65N30, 35L65
DOI: https://doi.org/10.1090/S0025-5718-10-02334-3
Published electronically: March 18, 2010
MathSciNet review: 2629996
Full-text PDF Free Access
View in AMS MathViewer New

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a new technique for the error analysis of hybridizable discontinuous Galerkin (HDG) methods. The technique relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods. This renders the analysis of the projections of the discretization errors simple and concise. By showing that these projections of the errors are bounded in terms of the distance between the solution and its projection, our studies of influence of the stabilization parameter are reduced to local analyses of approximation by the projection. We illustrate the technique on a specific HDG method applied to a model second-order elliptic problem.

1. D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, 7–32 (English, with French summary). MR 813687, https://doi.org/10.1051/m2an/1985190100071
2. 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, https://doi.org/10.1137/S0036142901384162
3. Jean-Pierre Aubin, Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Gelerkin’s and finite difference methods, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 21 (1967), 599–637. MR 233068
4. Peter Bastian and Béatrice Rivière, Superconvergence and 𝐻(𝑑𝑖𝑣) projection for discontinuous Galerkin methods, Internat. J. Numer. Methods Fluids 42 (2003), no. 10, 1043–1057. MR 1991232, https://doi.org/10.1002/fld.562
5. 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, https://doi.org/10.1007/BF01389710
6. Fatih Celiker and Bernardo Cockburn, Superconvergence of the numerical traces of discontinuous Galerkin and hybridized methods for convection-diffusion problems in one space dimension, Math. Comp. 76 (2007), no. 257, 67–96. MR 2261012, https://doi.org/10.1090/S0025-5718-06-01895-3
7. Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
8. Bernardo Cockburn and Bo Dong, An analysis of the minimal dissipation local discontinuous Galerkin method for convection-diffusion problems, J. Sci. Comput. 32 (2007), no. 2, 233–262. MR 2320571, https://doi.org/10.1007/s10915-007-9130-3
9. 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, https://doi.org/10.1090/S0025-5718-08-02123-6
10. 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, https://doi.org/10.1090/S0025-5718-05-01741-2
11. 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, https://doi.org/10.1137/070706616
12. 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, https://doi.org/10.1090/S0025-5718-08-02146-7
13. 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, https://doi.org/10.1137/S0036142900371544
14. Bernardo Cockburn, Guido Kanschat, and Dominik Schotzau, A locally conservative LDG method for the incompressible Navier-Stokes equations, Math. Comp. 74 (2005), no. 251, 1067–1095. MR 2136994, https://doi.org/10.1090/S0025-5718-04-01718-1
15. Lucia Gastaldi and Ricardo H. Nochetto, Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations, RAIRO Modél. Math. Anal. Numér. 23 (1989), no. 1, 103–128 (English, with French summary). MR 1015921, https://doi.org/10.1051/m2an/1989230101031
16. P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
17. J. Nitsche, Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens, Numer. Math. 11 (1968), 346–348 (German). MR 233502, https://doi.org/10.1007/BF02166687
18. 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) Springer, Berlin, 1977, pp. 292–315. Lecture Notes in Math., Vol. 606. MR 0483555
19. Rolf Stenberg, A family of mixed finite elements for the elasticity problem, Numer. Math. 53 (1988), no. 5, 513–538. MR 954768, https://doi.org/10.1007/BF01397550
20. Rolf Stenberg, Postprocessing schemes for some mixed finite elements, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 1, 151–167 (English, with French summary). MR 1086845, https://doi.org/10.1051/m2an/1991250101511

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65M60, 65N30, 35L65

Retrieve articles in all journals with MSC (2010): 65M60, 65N30, 35L65

Additional Information

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, Vincent Hall, 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

Francisco-Javier Sayas
Affiliation: Departamento de Matemática Aplicada, CPS, Universidad de Zaragoza, 50018 Zaragoza, Spain
Email: sayas002@umn.edu

DOI: https://doi.org/10.1090/S0025-5718-10-02334-3
Keywords: Discontinuous Galerkin methods, hybridization, superconvergence, postprocessing
Received by editor(s): December 22, 2008
Received by editor(s) in revised form: April 9, 2009
Published electronically: March 18, 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 partially supported by MEC/FEDER Project MTM2007–63204, Gobierno de Aragón (Grupo PDIE) and was a Visiting Professor of the School of Mathematics, University of Minnesota, during the development of this work.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.