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Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates. (English) Zbl 0567.65078

This paper is concerned with a technique for implementing certain mixed finite elements based on the use of Lagrange multipliers to impose interelement continuity. The matrices arising from this implementation are positive definite. Considering some well-known mixed methods, namely the Raviart-Thomas methods for second order elliptic problems and the Hellan-Herrmann-Johnson method for biharmonic problems, the authors show that the computed Lagrange multipliers may be exploited in a simple postprocess to produce better approximation of the original variables. Moreover, an equivalence between the mixed methods and certain modified versions of nonconforming methods is considered.
Reviewer: J.Lovíšek

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35J25 Boundary value problems for second-order elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
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References:

[1] I BABUSKA and J E OSBORN, Generalized finite element methods their performance and their relation to mixed methods, SIAM J Numer Anal 20 (1983), 510-536 Zbl0528.65046 MR701094 · Zbl 0528.65046
[2] I BABUSKA, J OSBORN and J PITKARANTA, Analysis of mixed methods using mesh dependent norms, Math Comput 35 (1980), 1039-1062 Zbl0472.65083 MR583486 · Zbl 0472.65083
[3] A BENSOUSSON, J L LIONS, G PAPANICOLAU, Asymptotic Analysis of Periodic Structures, North-Holland, Amsterdam, 1978 Zbl0404.35001 MR503330 · Zbl 0404.35001
[4] F BREZZI and P A RAVIART, Mixed finite element methods for 4th order elliptic equations, in Proc of the Royal Irish Academy Conference on Numerical Analysis, Academic Press, London, 1977 Zbl0434.65085 MR657975 · Zbl 0434.65085
[5] P G CIARLET, The Finite Element Method for Elliptic Equations, North-Holland, Amsterdam, 1978 MR520174 · Zbl 0383.65058
[6] J DOUGLAS and J E ROBERTS, Global estimates for mixed methods for second order elliptics, to appear in Math Comput Zbl0624.65109 · Zbl 0624.65109
[7] [7] R S FALK and J E OSBORN, Error estimates for mixed methods, R A I R O Anal numer 14 (1980), 309-324 Zbl0467.65062 MR592753 · Zbl 0467.65062
[8] B FRAEJIS DE VEUBEKE, Displacement and equilibrium models in the finite element method, in Stress Analysis, O C Zienkiewicz and G Holister, eds , Wiley, New York, 1965
[9] 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
[10] L HERRMANN, Finite element bending analysis for plates, J Eng Mech Div ASCE, a 3, EM5 (1967), 49-83
[11] [11] P LASCAUX and P LESAINT, Some nonconforming finite elements for the plate bending problem, R A I R O Anal numer 9 (1975), 9-53 Zbl0319.73042 MR423968 · Zbl 0319.73042
[12] [12] C JOHNSON, On the convergence of a mixed finite element method for plate bending problems, Numer Math 21 (1973), 43-62 Zbl0264.65070 MR388807 · Zbl 0264.65070
[13] L S D MORLEY, The triangular equilibrium element in the solution of plate bending problems, Aero Quart 19 (1968), 149-169
[14] [14] R RANNACHER, Nonconforming finite element methods for eigenvalue problems in linear plate theory, Numer Math 33 (1979), 23-42 Zbl0394.65035 MR545740 · Zbl 0394.65035
[15] [15] R RANNACHER, On nonconforming and mixed finite elements for plate bending problems The linear case R A I R O Anal numer 13 (1979), 369-387 Zbl0425.35042 MR555385 · Zbl 0425.35042
[16] P A RAVIART and J M THOMAS, A mixed finite element method for second order elliptic problems in Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics 606, Springer-Verlag, Berlin, 1977 Zbl0362.65089 MR483555 · Zbl 0362.65089
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