Analysis of some mixed finite element methods related to reduced integration

Authors:
Claes Johnson and Juhani Pitkäranta

Journal:
Math. Comp. **38** (1982), 375-400

MSC:
Primary 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1982-0645657-2

MathSciNet review:
645657

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove error estimates for the following two mixed finite element methods related to reduced integration: A method for Stokes' problem using rectangular elements with piecewise bilinear approximations for the velocities and piecewise constants for the pressure, and one method for a plate problem using bilinear approximations for transversal displacement and rotations and piecewise constants for the shear stress. The main idea of the proof in the case of Stokes' problem is to combine a weak Babuška-Brezzi type stability estimate for the pressure with a superapproximability property for the velocities. A similar technique is used in the case of the plate problem.

**[1]**Ivo Babuška,*Error-bounds for finite element method*, Numer. Math.**16**(1970/71), 322–333. MR**288971**, https://doi.org/10.1007/BF02165003**[2]**I. Babuška, J. Osborn & J. Pitkäranta,*Analysis of Mixed Methods Using Mesh Dependent Norms*, Report #2003, Mathematics Research Center, University of Wisconsin, 1979.**[3]**M. Bercovier,*Perturbation of mixed variational problems. Application to mixed finite element methods*, RAIRO Anal. Numér.**12**(1978), no. 3, 211–236, iii (English, with French summary). MR**509973**, https://doi.org/10.1051/m2an/1978120302111**[4]**F. Brezzi,*On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge**8**(1974), no. no. , no. R-2, 129–151 (English, with French summary). MR**365287****[5]**Philippe G. Ciarlet,*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174****[6]**M. Crouzeix and P.-A. Raviart,*Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge**7**(1973), no. no. , no. R-3, 33–75. MR**343661****[7]**V. Girault,*A combined finite element and marker and cell method for solving Navier-Stokes equations*, Numer. Math.**26**(1976), no. 1, 39–59. MR**449179**, https://doi.org/10.1007/BF01396565**[8]**V. Girault and P.-A. Raviart,*Finite element approximation of the Navier-Stokes equations*, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. MR**548867****[9]**R. Glowinski and O. Pironneau,*On numerical methods for the Stokes problem*, Energy methods in finite element analysis, Wiley, Chichester, 1979, pp. 243–264. MR**537009****[10]**V. A. Kondrat′ev,*Boundary value problems for elliptic equations in domains with conical or angular points*, Trudy Moskov. Mat. Obšč.**16**(1967), 209–292 (Russian). MR**0226187****[11]**D. Malkus & T. Hughes, "Mixed finite element methods--Reduced and selective integration techniques: A unification of concepts,"*Comput. Methods Appl. Mech. Engrg.*, v. 15, 1978, pp. 63-81.**[12]**H. Melzer & R. Rannacher,*Spannungskonzentrationen in Eckpunkten der vertikalen belasteten Kirchoffschen Platte*, Universität Bonn, 1979. (Preprint.)**[13]**R. L. Sani, P. M. Gresho & R. L. Lee,*On the Spurious Pressures Generated by Certain GFEM Solutions of the Incompressible Navier-Stokes Equations*, Technical report, Lawrence Livermore Laboratory, Oct. 1979.**[14]**Ranbir S. Sandhu and Kamar J. Singh,*Reduced integration for improved accuracy of finite element approximations*, Comput. Methods Appl. Mech. Engrg.**14**(1978), no. 1, 23–37. MR**495022**, https://doi.org/10.1016/0045-7825(78)90011-7**[15]**Roger Temam,*Une méthode d’approximation de la solution des équations de Navier-Stokes*, Bull. Soc. Math. France**96**(1968), 115–152 (French). MR**237972**

Retrieve articles in *Mathematics of Computation*
with MSC:
65N30

Retrieve articles in all journals with MSC: 65N30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1982-0645657-2

Article copyright:
© Copyright 1982
American Mathematical Society