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Superconvergence of mixed finite element methods for parabolic equations. (English) Zbl 0621.65116
The main objective of this paper is the establishment of superconvergence of the solution of a semidiscrete mixed finite element method to the solution of linear parabolic problems in $${\mathbb{R}}^ 2$$. A quasi- projection for mixed methods for linear parabolic problems is introduced and then used to produce asymptotic expansions to high order of the mixed method solution. Superconvergence is then derived by post-processing. Optimal order error estimates in Sobolev spaces of negative index are also shown.
Reviewer: N.F.F.Ebecken

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35K20 Initial-boundary value problems for second-order parabolic equations
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##### References:
 [1] D. N. ARNOLD and J. DOUGLAS Jr., Superconvergence of the Galerkin approximation of a quasilinear parabolic equation in a single space variable, Calcolo, 16 (1979), pp.345-369. Zbl0435.65094 MR592476 · Zbl 0435.65094 [2] [2] J. H. BRAMBLE and A. H. SCHATZ, Estimates for spline projections, RAIRO Anal. Numér., 8 (1976), pp. 5-37. Zbl0343.65045 MR436620 · Zbl 0343.65045 [3] J. H BRAMBLE and A. H. SCHATZ, Higher order local accuracy by averaging in the finite element method, Math. Comp. 137 (1977), pp. 94-111. Zbl0353.65064 MR431744 · Zbl 0353.65064 [4] J. DOUGLAS Jr., Superconvergence in the pressure in the simulation of miscible displacement, SIAM J. Numer. Anal., 22 (1985), pp.962-969. Zbl0624.65124 MR799123 · Zbl 0624.65124 [5] J. DOUGLAS Jr., T. DUPONT and M. F. WHEELER, A quasi-projection analysis of Galerkin methods for parabolic and hyperbolic equations, Math. Comp., 142 (1978), pp. 345-362. Zbl0385.65052 MR495012 · Zbl 0385.65052 [6] [6] J. DOUGLAS Jr., and F. A. MILNER, Interior and superconvergence estimates for mixed methods for second order elliptic problems, to Math. Modelling and Numer. Anal., 3 (1985), pp. 397-428. Zbl0613.65110 MR807324 · Zbl 0613.65110 [7] J. DOUGLAS Jr., and J. E. ROBERTS, Mixed finite element methods for second order elliptic problems, Mat. Apl. Comput., 1 (1982), pp.91-103. Zbl0482.65057 MR667620 · Zbl 0482.65057 [8] J. DOUGLAS Jr., and J. E. ROBERTS, Global estimates for mixed methods for second order elliptic equations, Math. Comp., 44 (1985), pp. 39-52. Zbl0624.65109 MR771029 · Zbl 0624.65109 [9] [9] R. FALK and J. OSBORN, Error estimates for mixed methods, RAIRO Anal. Numér., 14 (1980), pp. 249-277. Zbl0467.65062 MR592753 · Zbl 0467.65062 [10] [10] C. JOHNSON and V. THOMÉE, Error estimates for some mixed finite element methods for parabolic type problems, RAIRO Anal. Numér., 1 (1981), pp. 41-78. Zbl0476.65074 MR610597 · Zbl 0476.65074 [11] [11] J. C. NEDELEC, Mixed finite elements in R3, Numer. Math., 35 (1980), pp. 315-341. Zbl0419.65069 MR592160 · Zbl 0419.65069 [12] P. A. RAVI ART and J. M. THOMAS, A mixed finite element method for second order elliptic problems, in Proceedings of a conference on Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics 606, Springer-Verlag, Berlin, 1977, p. 292-315. Zbl0362.65089 MR483555 · Zbl 0362.65089 [13] M. C. SQUEFF, Superconvergence of Mixed Finite Element Methods for Parabolic Equation, Thesis, The University of Chicago, August 1985. [14] J. M. THOMAS, Sur l’analyse numérique des méthodes d’éléments finis hybrides et mixtes, Thèse, Université P. et M. Curie, Paris, 1977. [15] V. THOMÉE, Negative norm estimates and superconvergence in Galerkin methods for parabolic problems, Math. Comp., 34 (1980), pp.93-113. Zbl0454.65077 MR551292 · Zbl 0454.65077
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