Maximum norm estimates in the finite element method on plane polygonal domains. I

Schatz, A. H.; Wahlbin, L. B.

doi:10.1090/S0025-5718-1978-0502065-1

Maximum norm estimates in the finite element method on plane polygonal domains. I
HTML articles powered by AMS MathViewer

by A. H. Schatz and L. B. Wahlbin PDF

Math. Comp. 32 (1978), 73-109 Request permission

Abstract:

The finite element method is considered when applied to a model Dirichlet problem on a plane polygonal domain. Rate of convergence estimates in the maximum norm, up to the boundary, are given locally. The rate of convergence may vary from point to point and is shown to depend on the local smoothness of the solution and on a possible pollution effect. In one of the applications given, a method is proposed for calculating the first few coefficients (stress intensity factors) in an expansion of the solution in singular functions at a corner from the finite element solution. In a second application the location of the maximum error is determined. A rather general class of non-quasi-uniform meshes is allowed in our present investigations. In a subsequent paper, Part 2 of this work, we shall consider meshes that are refined in a systematic fashion near a corner and derive sharper results for that case.

References

Ivo Babuška, Finite element method for domains with corners, Computing (Arch. Elektron. Rechnen) 6 (1970), 264–273 (English, with German summary). MR 293858, DOI 10.1007/bf02238811
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
I. Babuška and A. K. Aziz, On the angle condition in the finite element method, SIAM J. Numer. Anal. 13 (1976), no. 2, 214–226. MR 455462, DOI 10.1137/0713021
Ivo Babuška and Michael B. Rosenzweig, A finite element scheme for domains with corners, Numer. Math. 20 (1972/73), 1–21. MR 323129, DOI 10.1007/BF01436639
James H. Bramble and Miloš Zlámal, Triangular elements in the finite element method, Math. Comp. 24 (1970), 809–820. MR 282540, DOI 10.1090/S0025-5718-1970-0282540-0
Philippe G. Ciarlet, Numerical analysis of the finite element method, Séminaire de Mathématiques Supérieures, No. 59 (Été 1975), Les Presses de l’Université de Montréal, Montreal, Que., 1976. MR 0495010
P. G. Ciarlet and P.-A. Raviart, General Lagrange and Hermite interpolation in $\textbf {R}^{n}$ with applications to finite element methods, Arch. Rational Mech. Anal. 46 (1972), 177–199. MR 336957, DOI 10.1007/BF00252458
Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
Jim Douglas Jr., Todd Dupont, and Mary Fanett Wheeler, An $L^{\infty }$ estimate and a superconvergence result for a Galerkin method for elliptic equations based on tensor products of piecewise polynomials, Rev. Française Automat. Informat. Recherche Opérationnelle Sér Rouge 8 (1974), no. R-2, 61–66 (English, with Loose French summary). MR 0359358
S. C. Eisenstat and M. H. Schultz, Computational aspects of the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 505–524. MR 0408269
J. Frehse and R. Rannacher, Eine $L^{1}$-Fehlerabschätzung für diskrete Grundlösungen in der Methode der finiten Elemente, Finite Elemente (Tagung, Univ. Bonn, Bonn, 1975) Bonn. Math. Schrift., No. 89, Inst. Angew. Math., Univ. Bonn, Bonn, 1976, pp. 92–114 (German, with English summary). MR 0471370

Proc. First Internat. Conf. on Structural Mech. in Reactor Technology

Pierre Grisvard, Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 207–274. MR 0466912
Stephen Hilbert, A mollifier useful for approximations in Sobolev spaces and some applications to approximating solutions of differential equations, Math. Comp. 27 (1973), 81–89. MR 331715, DOI 10.1090/S0025-5718-1973-0331715-3
Pierre Jamet, Estimations d’erreur pour des éléments finis droits presque dégénérés, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 10 (1976), no. R-1, 43–60 (French, with Loose English summary). MR 0455282
R. B. Kellogg, Higher order singularities for interface problems, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 589–602. MR 0433926, DOI 10.1007/bf01932971
V. A. Kondrat′ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč. 16 (1967), 209–292 (Russian). MR 0226187

Problèmes aux Limites Non Homogènes et Applications

Frank Natterer, Über die punktweise Konvergenz finiter Elemente, Numer. Math. 25 (1975/76), no. 1, 67–77 (German, with English summary). MR 474884, DOI 10.1007/BF01419529

Les Méthodes Directes en Théorie des Équations Elliptiques

Rev. Roumaine Math. Pures Appl.

J. A. Nitsche, $L_{\infty }$-convergence of finite element approximation, Journées “Éléments Finis” (Rennes, 1975) Univ. Rennes, Rennes, 1975, pp. 18. MR 568857

Conference on Finite Elements

J. Nitsche, On Dirichlet problems using subspaces with nearly zero boundary conditions, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 603–627. MR 0426456
Joachim A. Nitsche and Alfred H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974), 937–958. MR 373325, DOI 10.1090/S0025-5718-1974-0373325-9
J. Nitsche and A. Schatz, On local approximation properties of $L_{2}$-projection on spline-subspaces, Applicable Anal. 2 (1972), 161–168. MR 397268, DOI 10.1080/00036817208839035
L. A. Oganesjan and L. A. Ruhovec, Variational-difference schemes for second order linear elliptic equations in a two-dimensional region with a piecewise-smooth boundary, Ž. Vyčisl. Mat i Mat. Fiz. 8 (1968), 97–114 (Russian). MR 233525

Proc. World Congress on Finite Element Methods in Structural Mechanics

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 10.1090/S0025-5718-1977-0431753-X
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 10.1090/S0025-5718-1976-0436617-2

Amer. Math. Soc. Transl.

Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Gilbert Strang, Approximation in the finite element method, Numer. Math. 19 (1972), 81–98. MR 305547, DOI 10.1007/BF01395933
R. W. Thatcher, The use of infinite grid refinements at singularities in the solution of Laplace’s equation, Numer. Math. 25 (1975/76), no. 2, 163–178. MR 400748, DOI 10.1007/BF01462270
S. V. Uspenskiĭ, An imbedding theorem for S. L. Sobolev’s classes of fractional order $W_{p^{r}}$, Soviet Math. Dokl. 1 (1960), 132–133. MR 0124731