The Hellan-Herrmann-Johnson method: some new error estimates and postprocessing
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- by M. I. Comodi PDF
- Math. Comp. 52 (1989), 17-29 Request permission
Abstract:
We analyze the behavior of the mixed Hellan-Herrmann-Johnson method for solving the biharmonic problem ${\Delta ^2}\psi = f$. We show a superconvergence result for the distance between ${\psi ^h}$ (the approximation of the displacement) and ${P_h}\psi$ (where ${P_h}$ is a suitable projection operator). If the discrete equations are solved (as is usually done) by using interelement Lagrange multipliers, our superconvergence result allows us to prove the convergence, in suitable norms, of the Lagrange multipliers to the normal derivative of the displacement, and to construct a new approximation of $\nabla \psi$ which converges to $\nabla \psi$ faster than $\nabla {\psi ^h}$.References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, 7–32 (English, with French summary). MR 813687, DOI 10.1051/m2an/1985190100071
- I. Babuška, J. Osborn, and J. Pitkäranta, Analysis of mixed methods using mesh dependent norms, Math. Comp. 35 (1980), no. 152, 1039–1062. MR 583486, DOI 10.1090/S0025-5718-1980-0583486-7
- James H. Bramble and Jinchao Xu, A local post-processing technique for improving the accuracy in mixed finite-element approximations, SIAM J. Numer. Anal. 26 (1989), no. 6, 1267–1275. MR 1025087, DOI 10.1137/0726073
- Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. MR 799685, DOI 10.1007/BF01389710 F. Brezzi & M. Fortin, Mixed and Hybrid F.E.M. (To appear.)
- F. Brezzi and P.-A. Raviart, Mixed finite element methods for 4th order elliptic equations, Topics in numerical analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976) Academic Press, London, 1977, pp. 33–56. MR 0657975
- 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 C. Chinosi, L. Della Croce, L. D. Marini, A. Quarteroni, G. Sacchi & T. Scapolla, Implementation of Some Non Standard Finite Element Method for Fourth Order Problems, Report 231 of I.A.N.-C.N.R., Pavia, 1979.
- M. I. Comodi, Approximation of a bending plate problem with a boundary unilateral constraint, Numer. Math. 47 (1985), no. 3, 435–458. MR 808562, DOI 10.1007/BF01389591
- Jim Douglas Jr. and Jean E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), no. 169, 39–52. MR 771029, DOI 10.1090/S0025-5718-1985-0771029-9 B. Fraejis De Veubeke, "Displacement and equilibrium models in the finite element method," in Stress Analysis (O.C. Zienkiewicz and C. Holister, eds.), Wiley, New York, 1965. K. Hellan, Analysis of Elastic Plates in Flexure by a Simplified Finite Element Method, Acta Polytechnica Scandinavica, Ci 46, Trondheim, 1967. K. Herrmann, "Finite element bending analysis for plates," J. Eng. Mech. Div. ASCE, a3, EM5 93, 1967, pp. 49-83. L. Herrmann, A Bending Analysis for Plates, Proc. Conf. on Matrix Methods in Structural Mechanics, AFFDL-TR-66-88, 1965, 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 J. L. Lions & E. Magenes, Problèmes aux Limites non Homogènes et Applications, Tome I, Dunod, Paris, 1968. L. S. D. Morley, "The triangular equilibrium element in the solution of plate bending problems," Aero. Quart., v. 19, 1968, pp. 149-169. J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Academia, Prague, 1967.
- T. Scapolla, A mixed finite element method for the biharmonic problem, RAIRO Anal. Numér. 14 (1980), no. 1, 55–79 (English, with French summary). MR 566090
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Math. Comp. 52 (1989), 17-29
- MSC: Primary 65N30; Secondary 73C35
- DOI: https://doi.org/10.1090/S0025-5718-1989-0946601-7
- MathSciNet review: 946601