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Gastaldi, Lucia; Nochetto, Ricardo

Optimal \(L^{\infty}\)-error estimates for nonconforming and mixed finite element methods of lowest order. (English) Zbl 0597.65080

Numer. Math. 50, 587-611 (1987).
For second order linear elliptic problems it is proved that the \(P_ 1\)- nonconforming finite element method has the same \(L^{\infty}\)- asymptotic accuracy as the \(P_ 1\)-conforming one. This result is applied to derive optimal \(L^{\infty}\)-error estimates for both the displacement and the stress fields of the lowest order Raviart-Thomas mixed finite element method, and a superconvergence result at the barycenter of each element.

Cited in 19 Documents

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
74S05 Finite element methods applied to problems in solid mechanics

Keywords:

L-infinity error estimates; nonconforming finite element; displacement; stress fields; Raviart-Thomas; superconvergence
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\textit{L. Gastaldi} and \textit{R. Nochetto}, Numer. Math. 50, 587--611 (1987; Zbl 0597.65080)
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References:

[1] Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. M2 AN19, 7-35 (1985) · Zbl 0567.65078
[2] Baiocchi, C.: Estimation d’erreur dansL ? pour les in?quations ? obstacle. Mathematical Aspects of Finite Element Method, Lect. Notes Math.606, 27-34 (1977)
[3] Ciarlet, P.G.: The finite element method for elliptic problems. Amsterdam: North Holland 1978 · Zbl 0383.65058
[4] Crouzeix, M., Raviart, P.A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. RAIRO Anal. Numer.3, 33-76 (1973) · Zbl 0302.65087
[5] Douglas, J., Jr., Roberts, J.E.: Global estimates for mixed methods for second order elliptic equations. Math. Comput.44, 39-52 (1985) · Zbl 0624.65109
[6] Frehse, J., Rannacher, R.: EineL 1-Fehlerabsch?tzung f?r diskrete Grundl?sungen in der Methode der finite Elemente, pp. 92-114. N. 89 Bonn, Math. Schrift 1976
[7] Frehse, J., Rannacher, R.: AsymptoticL ?-error estimates for linear finite element approximations of quasilinear boundary value problems. SIAM J. Numer. Anal.15, 418-431 (1978) · Zbl 0386.65049
[8] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order Berlin, Heidelberg, New York: Springer 1977 · Zbl 0361.35003
[9] Grisvard, P.: Alternative de Fredholm relative au probl?me de Dirichlet dans un polygone ou un poly?dre. Boll. Unione Mat. Ital.5, 132-164 (1972) · Zbl 0277.35035
[10] Grisvard, P.: Behaviour of solutions of the an elliptic boundary value problem in a polygonal or polyhedral domain. In: Numerical Solution of Partial Differential Equations-III (B. Hubbard, ed.), pp. 207-274 New York: Academic Press 1976
[11] Johnson, C., Thomee, V.: Error estimates for some mixed finite element methods for parabolic type problems. RAIRO Anal. Numer.15, 41-78 (1981) · Zbl 0476.65074
[12] Lascaux, P., Lesaint, P.: Some nonconforming finite elements for the plate bending problem. RAIRO Anal. Numer.9, 9-53 (1975) · Zbl 0319.73042
[13] Marini, L.D.: An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method. SIAM J. Numer. Anal.22, 493-496 (1985) · Zbl 0573.65082
[14] Natterer, F.: ?ber die punktweise Konvergenz finiter Elemente. Numer. Math25, 67-78 (1975) · Zbl 0331.65073
[15] Necas, J.: Les m?thodes directes en th?orie des ?quations elliptiques. Prague: Masson, Paris et Academia 1967
[16] Nitsche, J.A.:L ?-convergence of finite element approximations. Mathematical Aspects of the Finite Element Methods, Lect. Notes Math.606, 261-274 (1977)
[17] Nitsche, J.A.: Schauder estimates for finite element approximations on second order elliptic boundary value problems. In: Proc. Special Year Numer. Anal. (Babuska, Lin, Osborn, eds.), pp. 290-343. Lectures Notes No. 20., Univ. of Maryland 1981
[18] Rannacher, R., Scott R.: Some optimal error, estimates for piecewise linear finite element approximations. Math. Comput.38, 437-445 (1982) · Zbl 0483.65007
[19] Raviart, P.A., Thomas, J.M.: A mixed finite element method for second order elliptic problems. Mathematical Aspects of the Finite Element Method, Lect. Notes Math.606, 292-315 (1977) · Zbl 0362.65089
[20] Schatz, A.H., Wahlbin, L.B.: On the quasi-optimality inL ? of theH 0 1 -projection into finite element spaces. Math. Comput.36, 1-22 (1982) · Zbl 0483.65006
[21] Scholz, R.:L ?-convergence of saddle-point approximations for second order problems. RAIRO Anal. Numer.11, 209-216 (1977) · Zbl 0356.35026
[22] Scholz, R.: OptimalL ?-estimates for a mixed finite element method for elliptic and parabolic problems. Calcolo20, 355-377 (1983) · Zbl 0571.65092
[23] Scott, R.: OptimalL ?-estimates for the finite element method on irregular meshes. Math. Comput.30, 681-697 (1976) · Zbl 0349.65060
[24] Strang, G.: Variational crimes in the finite element method. In: The mathematical foundations of the finite element method with applications to partial differential equations (A.K. Aziz, ed.), pp. 689-710. New York: Academic Press 1972
[25] Strang, G. Fix, G.J.: An analysis of the finite element method. Englewood Cliffs: Prentice Hall 1973 · Zbl 0356.65096
[26] Wahlbin, L.B.: Maximum norm error estimates in the finite element method with isoparametric quadratic elements and numerical integration. RAIRO Anal. Numer.,12, 173-202 (1978) · Zbl 0382.65057
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