Conditions for superconvergence of HDG methods for Stokes flow

Cockburn, Bernardo; Shi, Ke

doi:10.1090/S0025-5718-2012-02644-5

Conditions for superconvergence of HDG methods for Stokes flow
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by Bernardo Cockburn and Ke Shi PDF

Math. Comp. 82 (2013), 651-671 Request permission

Abstract:

We provide an a priori error analysis of a wide class of finite element methods for the Stokes equations. The methods are based on the velocity gradient-velocity-pressure formulation of the equations and include new and old mixed and hybridizable discontinuous Galerkin methods. We show how to reduce the error analysis to the verification of some properties of an elementwise-defined projection and of the local spaces defining the methods. We also show that the projection of the errors only depends on the approximation properties of the projection. We then provide sufficient conditions for the superconvergence of the projection of the error in the approximate velocity. We give many examples of these methods and show how to systematically construct them from similar methods for the diffusion equation.

References

Franco Brezzi, Jim Douglas Jr., Michel Fortin, and L. Donatella Marini, Efficient rectangular mixed finite elements in two and three space variables, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 4, 581–604 (English, with French summary). MR 921828, DOI 10.1051/m2an/1987210405811
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
Bernardo Cockburn, Jayadeep Gopalakrishnan, Ngoc Cuong Nguyen, Jaume Peraire, and Francisco-Javier Sayas, Analysis of HDG methods for Stokes flow, Math. Comp. 80 (2011), no. 274, 723–760. MR 2772094, DOI 10.1090/S0025-5718-2010-02410-X
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
B. Cockburn and F.J. Sayas, Divergence–conforming HDG methods for Stokes flow. Submitted.
Bernardo Cockburn, Weifeng Qiu, and Ke Shi, Conditions for superconvergence of HDG methods for second-order elliptic problems, Math. Comp. 81 (2012), no. 279, 1327–1353. MR 2904581, DOI 10.1090/S0025-5718-2011-02550-0
Monique Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations, SIAM J. Math. Anal. 20 (1989), no. 1, 74–97. MR 977489, DOI 10.1137/0520006
M. Farhloul and M. Fortin, A new mixed finite element for the Stokes and elasticity problems, SIAM J. Numer. Anal. 30 (1993), no. 4, 971–990. MR 1231323, DOI 10.1137/0730051
Mohamed Farhloul and Michel Fortin, A mixed finite element for the Stokes problem using quadrilateral elements, Adv. Comput. Math. 3 (1995), no. 1-2, 101–113. MR 1314904, DOI 10.1007/BF02431998
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
R. B. Kellogg and J. E. Osborn, A regularity result for the Stokes problem in a convex polygon, J. Functional Analysis 21 (1976), no. 4, 397–431. MR 0404849, DOI 10.1016/0022-1236(76)90035-5
J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
N.C. Nguyen and J. Peraire, Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics. J. Comp. Phy. 231 (2012), 5955–5988.
N. C. Nguyen, J. Peraire, and B. Cockburn, A hybridizable discontinuous Galerkin method for Stokes flow, Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 9-12, 582–597. MR 2796169, DOI 10.1016/j.cma.2009.10.007
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