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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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Analysis of mixed methods using mesh dependent norms
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by I. Babuška, J. Osborn and J. Pitkäranta PDF
Math. Comp. 35 (1980), 1039-1062 Request permission


This paper analyzes mixed methods for the biharmonic problem by means of new families of mesh dependent norms which are introduced and studied. More specifically, several mixed methods are shown to be stable with respect to these norms and, as a consequence, error estimates are obtained in a simple and direct manner.
  • Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/71), 322–333. MR 288971, DOI 10.1007/BF02165003
  • I. BABUŠKA & A. AZIZ, "Survey lectures on the mathematical foundations of the finite element method" in The Mathematical Foundations of the Finite Element Method with Application to Partial Differential Equations (A. K. Aziz, Ed.), Academic Press, New York, 1973, pp. 5-359.
  • I. Babuška and J. Osborn, Analysis of finite element methods for second order boundary value problems using mesh dependent norms, Numer. Math. 34 (1980), no. 1, 41–62. MR 560793, DOI 10.1007/BF01463997
  • J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 112–124. MR 263214, DOI 10.1137/0707006
  • 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
  • F. Brezzi, Sur la méthode des éléments finis hybrides pour le problème biharmonique, Numer. Math. 24 (1975), no. 2, 103–131 (French). MR 391538, DOI 10.1007/BF01400961
  • F. BREZZI & P. RAVIART, "Mixed finite element methods for 4th order elliptic equations," Topics in Numerical Analysis III (J. Miller, Ed.), Academic Press, New York, 1978.
  • 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
  • P. G. Ciarlet and P.-A. Raviart, A mixed finite element method for the biharmonic equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974) Publication No. 33, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 125–145. MR 0657977
  • Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
  • Jim Douglas Jr. and Todd Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Computing methods in applied sciences (Second Internat. Sympos., Versailles, 1975) Lecture Notes in Phys., Vol. 58, Springer, Berlin, 1976, pp. 207–216. MR 0440955
  • R. S. Falk and J. E. Osborn, Error estimates for mixed methods, RAIRO Anal. Numér. 14 (1980), no. 3, 249–277 (English, with French summary). MR 592753, DOI 10.1051/m2an/1980140302491
  • Michel Fortin, An analysis of the convergence of mixed finite element methods, RAIRO Anal. Numér. 11 (1977), no. 4, 341–354, iii (English, with French summary). MR 464543, DOI 10.1051/m2an/1977110403411
  • R. Glowinski, Approximations externes, par éléments finis de Lagrange d’ordre un et deux, du problème de Dirichlet pour l’opérateur biharmonique. Méthode itérative de résolution des problèmes approchés, Topics in numerical analysis (Proc. Roy. Irish Acad. Conf., University Coll., Dublin, 1972) Academic Press, London, 1973, pp. 123–171 (French). MR 0351120
  • L. HERRMANN, "Finite element bending analysis for plates," J. Eng. Mech., Div. ASCE EM5, v. 93, 1967, pp. 49-83. L. HERRMANN, "A bending analysis for plates," Proc. Conf. on Matrix Methods in Structural Mechanics, AFFDL-TR-66-88, pp. 577-604.
  • Claes Johnson, On the convergence of a mixed finite-element method for plate bending problems, Numer. Math. 21 (1973), 43–62. MR 388807, DOI 10.1007/BF01436186
  • R. B. Kellogg and J. E. Osborn, A regularity result for the Stokes problem in a convex polygon, J. Functional Analysis 21 (1976), no. 4, 397–431. MR 0404849, DOI 10.1016/0022-1236(76)90035-5
  • B. Mercier, Numerical solution of the biharmonic problem by mixed finite elements of class $C^{0}$, Boll. Un. Mat. Ital. (4) 10 (1974), 133–149 (English, with Italian summary). MR 0378442
  • Tetsuhiko Miyoshi, A finite element method for the solutions of fourth order partial differential equations, Kumamoto J. Sci. (Math.) 9 (1972/73), 87–116. MR 0386298
  • John E. Osborn, Analysis of mixed methods using mesh dependent spaces, Computational methods in nonlinear mechanics (Proc. Second Internat. Conf., Univ. Texas, Austin, Tex., 1979) North-Holland, Amsterdam-New York, 1980, pp. 361–377. MR 576914
  • Rolf Rannacher, On nonconforming and mixed finite element method for plate bending problems. The linear case, RAIRO Anal. Numér. 13 (1979), no. 4, 369–387 (English, with French summary). MR 555385, DOI 10.1051/m2an/1979130403691
  • Reinhard Scholz, Approximation von Sattelpunkten mit finiten Elementen, Finite Elemente (Tagung, Univ. Bonn, Bonn, 1975) Bonn. Math. Schrift., No. 89, Inst. Angew. Math., Univ. Bonn, Bonn, 1976, pp. 53–66 (German, with English summary). MR 0471377
  • Reinhard Scholz, A mixed method for 4th order problems using linear finite elements, RAIRO Anal. Numér. 12 (1978), no. 1, 85–90, iii (English, with French summary). MR 483557, DOI 10.1051/m2an/1978120100851
  • Reinhard Scholz, Interior error estimates for a mixed finite element method, Numer. Funct. Anal. Optim. 1 (1979), no. 4, 415–429. MR 538563, DOI 10.1080/01630567908816025
  • J. THOMAS, Sur l’Analyse Numérique des Méthodes d’Eléments Finis Hybrides et Mixtes, Thesis, Université P & M Curie, Paris, 1977.
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Math. Comp. 35 (1980), 1039-1062
  • MSC: Primary 65N30; Secondary 65N15, 73K10
  • DOI:
  • MathSciNet review: 583486