Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Analysis of some mixed finite element methods related to reduced integration
HTML articles powered by AMS MathViewer

by Claes Johnson and Juhani Pitkäranta PDF
Math. Comp. 38 (1982), 375-400 Request permission

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.
References
  • Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/71), 322–333. MR 288971, DOI 10.1007/BF02165003
  • I. Babuška, J. Osborn & J. Pitkäranta, Analysis of Mixed Methods Using Mesh Dependent Norms, Report #2003, Mathematics Research Center, University of Wisconsin, 1979.
  • 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, DOI 10.1051/m2an/1978120302111
  • 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. R-2, 129–151 (English, with French summary). MR 365287
  • Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
  • 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. R-3, 33–75. MR 343661
  • 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, DOI 10.1007/BF01396565
  • 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
  • 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
  • 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
  • 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. H. Melzer & R. Rannacher, Spannungskonzentrationen in Eckpunkten der vertikalen belasteten Kirchoffschen Platte, Universität Bonn, 1979. (Preprint.) 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.
  • 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, DOI 10.1016/0045-7825(78)90011-7
  • 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
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC: 65N30
  • Retrieve articles in all journals with MSC: 65N30
Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Math. Comp. 38 (1982), 375-400
  • MSC: Primary 65N30
  • DOI: https://doi.org/10.1090/S0025-5718-1982-0645657-2
  • MathSciNet review: 645657