Variational crimes and 𝐿^{∞} error estimates in the finite element method

Goldstein, Charles I.

doi:10.1090/S0025-5718-1980-0583491-0

Variational crimes and $L^{\infty }$ error estimates in the finite element method

Author: Charles I. Goldstein
Journal: Math. Comp. 35 (1980), 1131-1157
MSC: Primary 65N30; Secondary 65N15
DOI: https://doi.org/10.1090/S0025-5718-1980-0583491-0
MathSciNet review: 583491
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In order to numerically solve a second-order linear elliptic boundary value problem in a bounded domain, using the finite element method, it is often necessary in practice to violate certain assumptions of the standard variational formulation. Two of these "variational crimes" will be emphasized here and it will be shown that optimal ${L^\infty }$ error estimates still hold. The first "crime" occurs when a nonconforming finite element method is employed, so that smoothness requirements are violated at interelement boundaries. The second "crime" occurs when numerical integration is employed, so that the bilinear form is perturbed. In both cases, the "patch test" is crucial to the proof of ${L^\infty }$ estimates, just as it was in the case of mean-square estimates.

C. I. Goldstein and L. R. Scott, Optimal maximum norm error estimates for some finite element methods for treating the Dirichlet problem, Calcolo 20 (1983), no. 1, 1–52. MR 747006, DOI https://doi.org/10.1007/BF02575891
Ridgway Scott, Optimal $L^{\infty }$ estimates for the finite element method on irregular meshes, Math. Comp. 30 (1976), no. 136, 681–697. MR 436617, DOI https://doi.org/10.1090/S0025-5718-1976-0436617-2
J. A. Nitsche, $L_{\infty }$-convergence of finite element approximation, Journées “Éléments Finis” (Rennes, 1975) Univ. Rennes, Rennes, 1975, pp. 18. MR 568857
J. A. Nitsche, $L_{\infty }$-convergence of finite element approximation, Journées “Éléments Finis” (Rennes, 1975) Univ. Rennes, Rennes, 1975, pp. 18. MR 568857
Frank Natterer, Über die punktweise Konvergenz finiter Elemente, Numer. Math. 25 (1975/76), no. 1, 67–77 (German, with English summary). MR 474884, DOI https://doi.org/10.1007/BF01419529
Rolf Rannacher, Zur $L^{\infty }$-Konvergenz linearer finiter Elemente beim Dirichlet-Problem, Math. Z. 149 (1976), no. 1, 69–77 (German). MR 488859, DOI https://doi.org/10.1007/BF01301633
A. H. Schatz and L. B. Wahlbin, Interior maximum norm estimates for finite element methods, Math. Comp. 31 (1977), no. 138, 414–442. MR 431753, DOI https://doi.org/10.1090/S0025-5718-1977-0431753-X
A. H. Schatz and L. B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains. I, Math. Comp. 32 (1978), no. 141, 73–109. MR 502065, DOI https://doi.org/10.1090/S0025-5718-1978-0502065-1
A. H. Schatz and L. B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains. I, Math. Comp. 32 (1978), no. 141, 73–109. MR 502065, DOI https://doi.org/10.1090/S0025-5718-1978-0502065-1
Joachim A. Nitsche, $L_{\infty }$-error analysis for finite elements, Mathematics of finite elements and applications, III (Proc. Third MAFELAP Conf., Brunel Univ., Uxbridge, 1978) Academic Press, London-New York, 1979, pp. 173–186. MR 559297

P. G. C1ARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.

Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. MR 0443377
L. B. Wahlbin, Maximum norm error estimates in the finite element method with isoparametric quadratic elements and numerical integration, RAIRO Anal. Numér. 12 (1978), no. 2, 173–202, v (English, with French summary). MR 502070, DOI https://doi.org/10.1051/m2an/1978120201731
J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR 0350177

IU. P. KRASOVSKIĬ, "Isolation of singularities of the Green’s function," Math. USSR-Izv., v. 1, 1967, pp. 935-966.

A. K. Aziz (ed.), The mathematical foundations of the finite element method with applications to partial differential equations, Academic Press, New York-London, 1972. MR 0347104
M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33–75. MR 343661

R. TEMAM & F. THOMASSET, "Numerical solution of Navier-Stokes equation by a finite element method," Proc. Conf. on Numerical Methods in Fluid Mechanics, Rapallo, Italy, 1976.

Todd Dupont and Ridgway Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comp. 34 (1980), no. 150, 441–463. MR 559195, DOI https://doi.org/10.1090/S0025-5718-1980-0559195-7