## Uniform-in-time superconvergence of HDG methods for the heat equation

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- by Brandon Chabaud and Bernardo Cockburn PDF
- Math. Comp.
**81**(2012), 107-129 Request permission

## Abstract:

We prove that the superconvergence properties of the hybridizable discontinuous Galerkin method for second-order elliptic problems do hold uniformly in time for the semidiscretization by the same method of the heat equation provided the solution is smooth enough. Thus, if the approximations are piecewise polynomials of degree $k$, the approximation to the gradient converges with the rate $h^{k+1}$ for $k\ge 0$ and the $L^2$-projection of the error into a space of lower polynomial degree*superconverges*with the rate $\sqrt {\log (T/h^2)} h^{k+2}$ for $k\ge 1$ uniformly in time. As a consequence, an element-by-element postprocessing converges with the rate $\sqrt {\log (T/h^2)} h^{k+2}$ for $k\ge 1$ also uniformly in time. Similar results are proven for the Raviart-Thomas and the Brezzi-Douglas-Marini mixed methods.

## References

- J. H. Bramble and A. H. Schatz,
*Higher order local accuracy by averaging in the finite element method*, Math. Comp.**31**(1977), no. 137, 94–111. MR**431744**, DOI 10.1090/S0025-5718-1977-0431744-9 - 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 - Hongsen Chen, Richard Ewing, and Raytcho Lazarov,
*Superconvergence of mixed finite element methods for parabolic problems with nonsmooth initial data*, Numer. Math.**78**(1998), no. 4, 495–521. MR**1606312**, DOI 10.1007/s002110050323 - 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, 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 - Jim Douglas Jr., Todd Dupont, and Mary F. Wheeler,
*A quasi-projection analysis of Galerkin methods for parabolic and hyperbolic equations*, Math. Comp.**32**(1978), no. 142, 345–362. MR**495012**, DOI 10.1090/S0025-5718-1978-0495012-2 - Jim Douglas Jr.,
*Superconvergence in the pressure in the simulation of miscible displacement*, SIAM J. Numer. Anal.**22**(1985), no. 5, 962–969. MR**799123**, DOI 10.1137/0722058 - Ricardo Durán,
*Superconvergence for rectangular mixed finite elements*, Numer. Math.**58**(1990), no. 3, 287–298. MR**1075159**, DOI 10.1007/BF01385626 - Kenneth Eriksson and Claes Johnson,
*Adaptive finite element methods for parabolic problems. I. A linear model problem*, SIAM J. Numer. Anal.**28**(1991), no. 1, 43–77. MR**1083324**, DOI 10.1137/0728003 - R. E. Ewing and R. D. Lazarov,
*Superconvergence of the mixed finite element approximations of parabolic problems using rectangular finite elements*, East-West J. Numer. Math.**1**(1993), no. 3, 199–212. MR**1253635** - R. E. Ewing, R. D. Lazarov, and J. Wang,
*Superconvergence of the velocity along the Gauss lines in mixed finite element methods*, SIAM J. Numer. Anal.**28**(1991), no. 4, 1015–1029. MR**1111451**, DOI 10.1137/0728054 - 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 - Claes Johnson and Vidar Thomée,
*Error estimates for some mixed finite element methods for parabolic type problems*, RAIRO Anal. Numér.**15**(1981), no. 1, 41–78 (English, with French summary). MR**610597** - 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** - Maria Cristina J. Squeff,
*Superconvergence of mixed finite element methods for parabolic equations*, RAIRO Modél. Math. Anal. Numér.**21**(1987), no. 2, 327–352 (English, with French summary). MR**896246**, DOI 10.1051/m2an/1987210203271 - 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 - Vidar Thomée,
*Galerkin finite element methods for parabolic problems*, Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 1997. MR**1479170**, DOI 10.1007/978-3-662-03359-3 - Vidar Thomée, Jinchao Xu, and Nai Ying Zhang,
*Superconvergence of the gradient in piecewise linear finite-element approximation to a parabolic problem*, SIAM J. Numer. Anal.**26**(1989), no. 3, 553–573. MR**997656**, DOI 10.1137/0726033 - Madhusmita Tripathy and Rajen K. Sinha,
*Superconvergence of $H^1$-Galerkin mixed finite element methods for parabolic problems*, Appl. Anal.**88**(2009), no. 8, 1213–1231. MR**2568434**, DOI 10.1080/00036810903208163

## Additional Information

**Brandon Chabaud**- Affiliation: Department of Mathematics, Pennsylvania State University, University Park, State College, Pennsylvania 16802
- Email: chabaud@math.psu.edu
**Bernardo Cockburn**- Affiliation: School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, Minnesota 55455
- Email: cockburn@math.umn.edu
- Received by editor(s): January 27, 2010
- Received by editor(s) in revised form: July 26, 2010
- Published electronically: July 14, 2011
- Additional Notes: Part of this work was done when the first author was at the School of Mathematics, University of Minnesota.

The second author was partially supported by the National Science Foundation (Grant DMS-0712955) and by the Minnesota Supercomputing Institute. Part of this work was done when this author was visiting the Research Institute for Mathematical Sciences, Kyoto University, Japan, during the Fall of 2009. - © Copyright 2011 American Mathematical Society
- Journal: Math. Comp.
**81**(2012), 107-129 - MSC (2010): Primary 65M60, 35K05
- DOI: https://doi.org/10.1090/S0025-5718-2011-02525-1
- MathSciNet review: 2833489