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A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. (English) Zbl 1198.65193

Summary: 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 \( k\geq0\) 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.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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[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). · Zbl 0567.65078
[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. · Zbl 1008.65080
[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. · Zbl 0688.65061
[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. · Zbl 0599.65072
[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. · Zbl 0987.65111
[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. · Zbl 1109.65078
[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. · Zbl 0932.65126
[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. · Zbl 0383.65058
[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. · Zbl 1143.76031
[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. · Zbl 1084.65113
[11] B. Cockburn, J. Gopalakrishnan, and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems, Submitted. · Zbl 1205.65312
[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. · Zbl 0927.65118
[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. · Zbl 0624.65109
[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). · Zbl 0673.65060
[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.
[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.
[17] Rolf Stenberg, A family of mixed finite elements for the elasticity problem, Numer. Math. 53 (1988), no. 5, 513 – 538. · Zbl 0632.73063
[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). · Zbl 0717.65081
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