×

zbMATH — the first resource for mathematics

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

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
PDF BibTeX XML Cite
Full Text: DOI EuDML
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) · doi:10.1007/BFb0064453
[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 · doi:10.1090/S0025-5718-1985-0771029-9
[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 · doi:10.1137/0715026
[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 · doi:10.1137/0722029
[14] Natterer, F.: ?ber die punktweise Konvergenz finiter Elemente. Numer. Math25, 67-78 (1975) · Zbl 0331.65073 · doi:10.1007/BF01419529
[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) · doi:10.1007/BFb0064468
[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 · doi:10.1090/S0025-5718-1982-0645661-4
[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 · doi:10.1007/BFb0064470
[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 · doi:10.1007/BF02576470
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.