Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Analysis of mixed methods using mesh dependent norms


Authors: I. Babuška, J. Osborn and J. Pitkäranta
Journal: Math. Comp. 35 (1980), 1039-1062
MSC: Primary 65N30; Secondary 65N15, 73K10
MathSciNet review: 583486
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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.


References [Enhancements On Off] (What's this?)

  • [1] Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/1971), 322–333. MR 0288971 (44 #6166)
  • [2] 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.
  • [3] 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 (81g:65143), http://dx.doi.org/10.1007/BF01463997
  • [4] 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 0263214 (41 #7819)
  • [5] 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 loose French summary). MR 0365287 (51 #1540)
  • [6] F. Brezzi, Sur la méthode des éléments finis hybrides pour le problème biharmonique, Numer. Math. 24 (1975), no. 2, 103–131 (French). MR 0391538 (52 #12359)
  • [7] 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.
  • [8] 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 (58 #25001)
  • [9] 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), Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 125–145. Publication No. 33. MR 0657977 (58 #31907)
  • [10] Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. \jname RAIRO Analyse Numérique 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739 (53 #4569)
  • [11] 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) Springer, Berlin, 1976, pp. 207–216. Lecture Notes in Phys., Vol. 58. MR 0440955 (55 #13823)
  • [12] 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 (82j:65076)
  • [13] 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 0464543 (57 #4473)
  • [14] 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 (50 #3609)
  • [15] L. HERRMANN, "Finite element bending analysis for plates," J. Eng. Mech., Div. ASCE EM5, v. 93, 1967, pp. 49-83.
  • [16] L. HERRMANN, "A bending analysis for plates," Proc. Conf. on Matrix Methods in Structural Mechanics, AFFDL-TR-66-88, pp. 577-604.
  • [17] Claes Johnson, On the convergence of a mixed finite-element method for plate bending problems, Numer. Math. 21 (1973), 43–62. MR 0388807 (52 #9641)
  • [18] 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 (53 #8649)
  • [19] B. Mercier, Numerical solution of the biharmonic problem by mixed finite elements of class 𝐶⁰, Boll. Un. Mat. Ital. (4) 10 (1974), 133–149 (English, with Italian summary). MR 0378442 (51 #14610)
  • [20] 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 (52 #7156)
  • [21] 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 (81h:65116)
  • [22] 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 (80i:65125)
  • [23] Reinhard Scholz, Approximation von Sattelpunkten mit finiten Elementen, Finite Elemente (Tagung, Univ. Bonn, Bonn, 1975) Inst. Angew. Math., Univ. Bonn, Bonn, 1976, pp. 53–66. Bonn. Math. Schrift., No. 89 (German, with English summary). MR 0471377 (57 #11111)
  • [24] 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 0483557 (58 #3549)
  • [25] Reinhard Scholz, Interior error estimates for a mixed finite element method, Numer. Funct. Anal. Optim. 1 (1979), no. 4, 415–429. MR 538563 (80g:65116), http://dx.doi.org/10.1080/01630567908816025
  • [26] J. THOMAS, Sur l'Analyse Numérique des Méthodes d'Eléments Finis Hybrides et Mixtes, Thesis, Université P & M Curie, Paris, 1977.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30, 65N15, 73K10

Retrieve articles in all journals with MSC: 65N30, 65N15, 73K10


Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1980-0583486-7
PII: S 0025-5718(1980)0583486-7
Keywords: Mixed methods, error estimates, stability
Article copyright: © Copyright 1980 American Mathematical Society