Postprocessing schemes for some mixed finite elements. (English) Zbl 0717.65081

The author discusses some mixed finite element approximations of two model problems i.e. Poisson’s equation and the biharmonic equation. Some new postprocessing schemes are presented and performed separately on each element so that one can obtain a considerably better approximation for the scalar variable than the original one.
Reviewer: P.K.Mahanti


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J40 Boundary value problems for higher-order elliptic equations
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
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[1] [1] D. N. ARNOLD and F. BREZZI, Mixed and Nonconforming Finite Element Methods : Implementation, Postprocessing and Error Estimates, RAIRO, M2AN, Vol. 19, 1985, pp. 7-32. Zbl0567.65078 MR813687 · Zbl 0567.65078
[2] [2] F. BREZZI, J. DOUGLAS, R. DURÁN and M. FORTIN, Mixed Finite Elements for Second Order Elliptic Problems in Three Variables, Numer. Math., Vol.51, 1987, pp. 237-250. Zbl0631.65107 MR890035 · Zbl 0631.65107 · doi:10.1007/BF01396752
[3] I. BABUŠKA, J. E. OSBORN and J. PITKÄRANTA, Analysis of Mixed Methods using Mesh Dependent Norms, Math. Comp., Vol 35, 1980, pp. 1039-1062. Zbl0472.65083 MR583486 · Zbl 0472.65083 · doi:10.2307/2006374
[4] F. BREZZI, J. DOUGLAS and L. D. MARINI, Two Families of Mixed Finite Elements for Second Order Elliptic Equations, Numer. Math., Vol.47, 1985, pp. 19-34. Zbl0599.65072 MR808322 · Zbl 0599.65072 · doi:10.1007/BF01389710
[5] P. G. ClARLET, The Finite Element Method for Elliptic Problems, North-Holland, 1978. Zbl0383.65058 MR520174 · Zbl 0383.65058
[6] M. I. COMODI, The Hellan-Herrmann-Johnson Method: Estimates for the Lagrange Multiplier and Postprocessing, Math. Comp. Vol. 52, 1989, pp. 17-29. Zbl0665.65082 MR946601 · Zbl 0665.65082 · doi:10.2307/2008650
[7] J. DOUGLAS and J. E. SANTOS, Approximation of Waves in Composite Media, The Mathematics of Finite Elements and Applications VI. MAFELAP 1987, J. R. Whiteman (Ed.), Academic Press, 55-74. Zbl0682.73028 MR956888 · Zbl 0682.73028
[8] [8] R. S. FALK and J. E. OSBORN, Error Estimates for Mixed Methods, RAIRO,Anal. Numer., Vol 14, 1980, pp. 249-278. Zbl0467.65062 MR592753 · Zbl 0467.65062
[9] V. GlRAULT and P. A. RAVIART, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer 1986. Zbl0585.65077 MR540128 · Zbl 0585.65077
[10] K. HELLAN, Analysis of Elastic Plates in Flexure by a Simplified Finite Element Method, Acta Polytechnica Scandinavica, Ci 46, Trondheim, 1967. Zbl0237.73046 · Zbl 0237.73046
[11] L. HERRMANN, Finite Element Bending Analysis for Plates, J. Eng. Div. ASCE, a3, EM5, 1967, pp. 49-83.
[12] [12] C. JOHNSON, On the Convergence of a Mixed Finite Method for Plate Bending Problems, Numer. Math. Vol. 21, 1973, pp. 43-62. Zbl0264.65070 MR388807 · Zbl 0264.65070 · doi:10.1007/BF01436186
[13] L. D. MARINI and A. SAVINI, Accurate Computation of Electric Field in Reverse Biased Semiconductor Devices. A Mixed Finite Element Approach, Compel, Vol. 3, 1984, pp. 123-135. Zbl0619.65120 · Zbl 0619.65120 · doi:10.1108/eb009991
[14] [14] J. C. NEDELEC, Mixed Finite Elements in R3, Numer. Math., Vol. 35, 1980, pp. 315-341. Zbl0419.65069 MR592160 · Zbl 0419.65069 · doi:10.1007/BF01396415
[15] P. A. RAVIART and J. M. THOMAS, A Mixed Finite Element Method for 2nd Order Elliptic Problems, Proceedings of the Symposium on the Mathematical Aspects of the Finite Element Method. Lecture Notes in Mathematics 606, Springer 1977, pp. 292-315. Zbl0362.65089 MR483555 · Zbl 0362.65089
[16] [16] R. STENBERG, On the Construction of Optimal Mixed Finite Element Methods for the Linear Elasticity Problem, Numer. Math., Vol. 48, 1986, pp. 447-462. Zbl0563.65072 MR834332 · Zbl 0563.65072 · doi:10.1007/BF01389651
[17] R. STENBERG, On the Postprocessing of Mixed Equilibrium Finite Element Methods, Numerical Techniques in Continuüm Mechanics. Proceedings of the Second GAMM-Seminar, Kiel, January 17 to 19, 1986. W. Hackbusch, K. Witsch (Eds.), Vieweg, Braunschweig 1987, pp. 102-109. Zbl0645.73033 · Zbl 0645.73033
[18] M. WHEELER and R. GONZALES, Mixed Finite Element Methods for Petroleum Reservoir Simulation, Computing Methods in Applied Sciences and Engineering, VI, R. Glowinski, J. L. Lions (Eds.), North-Holland 1984, pp. 639-658. Zbl0598.76102 · Zbl 0598.76102
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