The HellanHerrmannJohnson method: some new error estimates and postprocessing
Author:
M. I. Comodi
Journal:
Math. Comp. 52 (1989), 1729
MSC:
Primary 65N30; Secondary 73C35
MathSciNet review:
946601
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Abstract: We analyze the behavior of the mixed HellanHerrmannJohnson method for solving the biharmonic problem . We show a superconvergence result for the distance between (the approximation of the displacement) and (where 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 which converges to faster than .
 [1]
Robert
A. Adams, Sobolev spaces, Academic Press [A subsidiary of
Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1975. Pure and
Applied Mathematics, Vol. 65. MR 0450957
(56 #9247)
 [2]
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
(87g:65126)
 [3]
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
(81m:65166), http://dx.doi.org/10.1090/S00255718198005834867
 [4]
James
H. Bramble and Jinchao
Xu, A local postprocessing technique for improving the accuracy in
mixed finiteelement approximations, SIAM J. Numer. Anal.
26 (1989), no. 6, 1267–1275. MR 1025087
(90m:65193), http://dx.doi.org/10.1137/0726073
 [5]
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
(87g:65133), http://dx.doi.org/10.1007/BF01389710
 [6]
F. Brezzi & M. Fortin, Mixed and Hybrid F.E.M. (To appear.)
 [7]
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
(58 #31905)
 [8]
Philippe
G. Ciarlet, The finite element method for elliptic problems,
NorthHolland Publishing Co., AmsterdamNew YorkOxford, 1978. Studies in
Mathematics and its Applications, Vol. 4. MR 0520174
(58 #25001)
 [9]
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.
 [10]
M.
I. Comodi, Approximation of a bending plate problem with a boundary
unilateral constraint, Numer. Math. 47 (1985),
no. 3, 435–458. MR 808562
(86k:65109), http://dx.doi.org/10.1007/BF01389591
 [11]
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
(86b:65122), http://dx.doi.org/10.1090/S00255718198507710299
 [12]
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.
 [13]
K. Hellan, Analysis of Elastic Plates in Flexure by a Simplified Finite Element Method, Acta Polytechnica Scandinavica, Ci 46, Trondheim, 1967.
 [14]
K. Herrmann, "Finite element bending analysis for plates," J. Eng. Mech. Div. ASCE, a3, EM5 93, 1967, pp. 4983.
 [15]
L. Herrmann, A Bending Analysis for Plates, Proc. Conf. on Matrix Methods in Structural Mechanics, AFFDLTR6688, 1965, pp. 577604.
 [16]
Claes
Johnson, On the convergence of a mixed finiteelement method for
plate bending problems, Numer. Math. 21 (1973),
43–62. MR
0388807 (52 #9641)
 [17]
J. L. Lions & E. Magenes, Problèmes aux Limites non Homogènes et Applications, Tome I, Dunod, Paris, 1968.
 [18]
L. S. D. Morley, "The triangular equilibrium element in the solution of plate bending problems," Aero. Quart., v. 19, 1968, pp. 149169.
 [19]
J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Academia, Prague, 1967.
 [20]
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
(81m:65178)
 [1]
 R. Adams, Sobolev Spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
 [2]
 D. N. Arnold & F. Brezzi, "Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates," RAIRO Modél. Math. Anal. Numér., v. 19, 1985, 732. MR 813687 (87g:65126)
 [3]
 I. Babuška, J. Osborn & J. Pitkäranta, "Analysis of mixed methods using mesh dependent norms," Math. Comp., v. 35, 1980, pp. 10391062. MR 583486 (81m:65166)
 [4]
 J. H. Bramble & J. Xu, "A local postprocessing technique for improving the accuracy in mixed finite element approximation," SIAM J. Numer. Anal. (To appear.) MR 1025087 (90m:65193)
 [5]
 F. Brezzi, J. Douglas, Jr. & L. D. Marini, "Two families of mixed finite elements for second order elliptic problems," Numer. Math., v. 47, 1985, pp. 217435. MR 799685 (87g:65133)
 [6]
 F. Brezzi & M. Fortin, Mixed and Hybrid F.E.M. (To appear.)
 [7]
 F. Brezzi & P. A. Raviart, "Mixed finite element methods for 4th order elliptic equations," in Topics in Numerical Analysis, Vol. III (J.J.H. Miller, ed.), Academic Press, London, 1977, pp. 3356. MR 0657975 (58:31905)
 [8]
 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
 [9]
 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.
 [10]
 M. I. Comodi, "Approximation of a bending plate problem with a boundary unilateral constraint," Numer. Math., v. 47, 1985, pp. 435458. MR 808562 (86k:65109)
 [11]
 J. Douglas, Jr. & J. E. Roberts, "Global estimates for mixed methods for second order elliptic equations," Math. Comp., v. 44, 1985, pp. 3952. MR 771029 (86b:65122)
 [12]
 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.
 [13]
 K. Hellan, Analysis of Elastic Plates in Flexure by a Simplified Finite Element Method, Acta Polytechnica Scandinavica, Ci 46, Trondheim, 1967.
 [14]
 K. Herrmann, "Finite element bending analysis for plates," J. Eng. Mech. Div. ASCE, a3, EM5 93, 1967, pp. 4983.
 [15]
 L. Herrmann, A Bending Analysis for Plates, Proc. Conf. on Matrix Methods in Structural Mechanics, AFFDLTR6688, 1965, pp. 577604.
 [16]
 C. Johnson, "On the convergence of a mixed finite element method for plate bending problems," Numer. Math., v. 21, 1973, pp. 4362. MR 0388807 (52:9641)
 [17]
 J. L. Lions & E. Magenes, Problèmes aux Limites non Homogènes et Applications, Tome I, Dunod, Paris, 1968.
 [18]
 L. S. D. Morley, "The triangular equilibrium element in the solution of plate bending problems," Aero. Quart., v. 19, 1968, pp. 149169.
 [19]
 J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Academia, Prague, 1967.
 [20]
 T. Scapolla, "A mixed finite element method for the biharmonic problem," RAIRO Anal. Numér., v. 14, 1980, pp. 5579. MR 566090 (81m:65178)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909466017
PII:
S 00255718(1989)09466017
Article copyright:
© Copyright 1989
American Mathematical Society
