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

Authors:
Bernardo Cockburn, Bo Dong and Johnny Guzmán

Journal:
Math. Comp. **77** (2008), 1887-1916

MSC (2000):
Primary 65M60, 65N30, 35L65

Published electronically:
May 6, 2008

MathSciNet review:
2429868

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 for both the potential as well as the flux, the order of convergence in of both unknowns is . Moreover, both the approximate potential as well as its numerical trace superconverge in -like norms, to suitably chosen projections of the potential, with order . This allows the application of element-by-element postprocessing of the approximate solution which provides an approximation of the potential converging with order in . 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.

**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****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**, 10.1137/S0036142901384162**3.**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**, 10.1137/0726073**4.**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**, 10.1007/BF01389710**5.**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 (electronic). MR**1813251**, 10.1137/S0036142900371003**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**, 10.1090/S0025-5718-06-01895-3**7.**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****8.**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****9.**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**, 10.1007/s10915-007-9130-3**10.**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 (electronic). MR**2051067**, 10.1137/S0036142902417893**11.**B. Cockburn, J. Gopalakrishnan, and R. Lazarov,*Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems*, Submitted.**12.**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 (electronic). MR**1655854**, 10.1137/S0036142997316712**13.**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**, 10.1090/S0025-5718-1985-0771029-9**14.**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****15.**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) Springer, Berlin, 1977, pp. 275-291. Lecture Notes in Math., Vol. 606. MR**0520341****16.**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****17.**Rolf Stenberg,*A family of mixed finite elements for the elasticity problem*, Numer. Math.**53**(1988), no. 5, 513–538. MR**954768**, 10.1007/BF01397550**18.**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**

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
65M60,
65N30,
35L65

Retrieve articles in all journals with MSC (2000): 65M60, 65N30, 35L65

Additional Information

**Bernardo Cockburn**

Affiliation:
School of Mathematics, 127 Vincent Hall, University of Minnesota, Minneapolis, Minnesota 55455

Email:
cockburn@math.umn.edu

**Bo Dong**

Affiliation:
Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912

Email:
bdong@dam.brown.edu

**Johnny Guzmán**

Affiliation:
School of Mathematics, 127 Vincent Hall, University of Minnesota, Minneapolis, Minnesota 55455

Email:
guzma033@umn.edu

DOI:
https://doi.org/10.1090/S0025-5718-08-02123-6

Keywords:
Discontinuous Galerkin methods,
hybridization,
superconvergence,
second-order elliptic problems

Received by editor(s):
November 1, 2006

Received by editor(s) in revised form:
September 6, 2007

Published electronically:
May 6, 2008

Additional Notes:
The first author was supported in part by the National Science Foundation (Grant DMS-0411254) and by the University of Minnesota Supercomputing Institute.

The third author was supported by an NSF Mathematical Science Postdoctoral Research Fellowship (DMS-0503050)

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.