A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems

Cockburn, Bernardo; Dong, Bo; Guzmán, Johnny

doi:10.1090/S0025-5718-08-02123-6

A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems
HTML articles powered by AMS MathViewer

by Bernardo Cockburn, Bo Dong and Johnny Guzmán HTML | PDF

Math. Comp. 77 (2008), 1887-1916 Request permission

Abstract:

We identify and study an LDG-hybridizable Galerkin method, which is not an LDG method, for second-order elliptic problems in several space dimensions with remarkable convergence properties. Unlike all other known discontinuous Galerkin methods using polynomials of degree $k\ge 0$ for both the potential as well as the flux, the order of convergence in $L^2$ of both unknowns is $k+1$. Moreover, both the approximate potential as well as its numerical trace superconverge in $L^2$-like norms, to suitably chosen projections of the potential, with order $k+2$. This allows the application of element-by-element postprocessing of the approximate solution which provides an approximation of the potential converging with order $k+2$ in $L^2$. The method can be thought to be in between the hybridized version of the Raviart-Thomas and that of the Brezzi-Douglas-Marini mixed methods.

References

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, DOI 10.1051/m2an/1985190100071
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
James H. Bramble and Jinchao Xu, A local post-processing technique for improving the accuracy in mixed finite-element approximations, SIAM J. Numer. Anal. 26 (1989), no. 6, 1267–1275. MR 1025087, DOI 10.1137/0726073
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
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
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, DOI 10.1090/S0025-5718-06-01895-3
Zhangxin Chen, Equivalence between and multigrid algorithms for nonconforming and mixed methods for second-order elliptic problems, East-West J. Numer. Math. 4 (1996), no. 1, 1–33. MR 1393063
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
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, DOI 10.1007/s10915-007-9130-3
Bernardo Cockburn and Jayadeep Gopalakrishnan, A characterization of hybridized mixed methods for second order elliptic problems, SIAM J. Numer. Anal. 42 (2004), no. 1, 283–301. MR 2051067, DOI 10.1137/S0036142902417893
B. Cockburn, J. Gopalakrishnan, and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems, Submitted.
Bernardo Cockburn and Chi-Wang Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998), no. 6, 2440–2463. MR 1655854, DOI 10.1137/S0036142997316712
Jim Douglas Jr. and Jean E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), no. 169, 39–52. MR 771029, DOI 10.1090/S0025-5718-1985-0771029-9
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, DOI 10.1051/m2an/1989230101031
J. T. Oden and J. K. Lee, Dual-mixed hybrid finite element method for second-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. 275–291. MR 0520341
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
Rolf Stenberg, A family of mixed finite elements for the elasticity problem, Numer. Math. 53 (1988), no. 5, 513–538. MR 954768, DOI 10.1007/BF01397550
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, DOI 10.1051/m2an/1991250101511