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

Authors:
A. H. Schatz and L. B. Wahlbin

Journal:
Math. Comp. **32** (1978), 73-109

MSC:
Primary 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1978-0502065-1

MathSciNet review:
0502065

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**I. BABUŠKA, "Finite element method for domains with corners,"*Computing*, v. 6, 1970, pp. 264-273. MR**0293858 (45:2934)****[2]**I. BABUŠKA & A. K. AZIZ, "Survey lectures on the mathematical foundations of the finite element method,"*The Mathematical Foundations of the Finite Element Method*(A. K. Aziz, Editor), Academic Press, New York, 1972. MR**0347104 (49:11824)****[3]**I. BABUŠKA & A. K. AZIZ, "On the angle condition in the finite element method,"*SIAM J. Numer. Anal.*, v. 13, 1976, pp. 214-226. MR**0455462 (56:13700)****[4]**I. BABUŠKA & W. RHEINBOLDT, "Mathematical problems of computational decisions in the finite element method," Technical Note TR-426, University of Maryland, 1975.**[5]**I. BABUŠKA & M. ROSENZWEIG, "A finite element scheme for domains with corners,"*Numer. Math.*, v. 20, 1973, pp. 1-21. MR**0323129 (48:1487)****[6]**J. BRAMBLE & M. ZLÁMAL, "Triangular elements in the finite element method,"*Math. Comp.*, v. 24, 1970, pp. 809-820. MR**0282540 (43:8250)****[7]**P. G. CIARLET,*Numerical Analysis of the Finite Element Method*, Séminaire de Mathématiques Supérieures, Presse de l'Université de Montréal, 1976. MR**0495010 (58:13778)****[8]**P. G. CIARLET & P.-A. RAVIART, "General Lagrange and Hermite interpolation in with applications to finite element methods,"*Arch. Rational Mech. Anal.*, v. 46, 1972, pp. 177-199. MR**0336957 (49:1730)****[9]**Ph. CLEMENT, "Approximation by finite element functions using local regularization,"*Rev. Française Automat. Informat. Recherche Opérationlle Sér. Rouge*, v. 9, 1975, pp. 77-84. MR**0400739 (53:4569)****[10]**J. DOUGLAS, JR., T. DUPONT & M. F. WHEELER, "An estimate and a superconvergence result for a Galerkin method based on tensor products of piecewise polynomials,"*Rev. Française Automat. Informat. Recherche Opérationelle Sér. Rouge*, v. 8, 1974, pp. 61-66. MR**0359358 (50:11812)****[11]**S. C. EISENSTAT & M. H. SCHULTZ, "Computational aspects of the finite element method,"*The Mathematical Foundations of the Finite Element Method*(A. K. Aziz, Editor), Academic Press, New York, 1972, pp. 505-524. MR**0408269 (53:12034)****[12]**J. FREHSE & R. RANNACHER, "Eine -Fehlerabschätzung diskreter Grundlösungen in der Methode der finiten Elemente, Tagungsband "Finite Elemente","*Bonn. Math. Schr.*, 1976. MR**0471370 (57:11104)****[13]**R. GALLAGHER, "Survey and evaluation of the finite element method in fracture mechanics analysis,"*Proc. First Internat. Conf. on Structural Mech. in Reactor Technology*, Berlin, vol. 6, part L, pp. 637-653.**[14]**P. GRISVARD, "Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain,"*Numerical Solution of Partial Differential Equations*-III (B. Hubbard, Editor), Academic Press, New York, 1976, pp. 207-274. MR**0466912 (57:6786)****[15]**S. HILBERT, "A mollifier useful for approximations in Sobolev spaces and some applications to approximating solutions of differential equations,"*Math. Comp.*, v. 27, 1973, pp. 81-89. MR**0331715 (48:10047)****[16]**P. JAMET, "Estimations d'erreur pour des éléments finis droits presque dégénérés,"*Rev. Française Automat. Informat. Recherche Opérationelle Sér. Rouge*, v. 10, 1976, pp. 43-61. MR**0455282 (56:13521)****[17]**R. B. KELLOGG, "Higher order singularities for interface problems,"*The Mathematical Foundations of the Finite Element Method*(A. K. Aziz, Editor), Academic Press, New York, 1972, pp. 589-602. MR**0433926 (55:6896)****[18]**R. B. KELLOGG, "Interpolation between subspaces of a Hilbert space," Technical Note BN-719, University of Maryland, 1971.**[19]**V. A. KONDRAT'EV, "Boundary problems for elliptic equations in domains with conical or angular points,"*Trans. Moscow Math. Soc.*, v. 16, 1967, pp. 227-313. MR**0226187 (37:1777)****[20]**J. L. LIONS & E. MAGENES,*Problèmes aux Limites Non Homogènes et Applications*, I, Dunod, Paris, 1968.**[21]**F. NATTERER, "Über die Punktweise Konvergenz Finiter Elemente,"*Numer. Math.*, v. 25, 1975, pp. 67-77. MR**0474884 (57:14514)****[22]**J. NEČAS,*Les Méthodes Directes en Théorie des Équations Elliptiques*, Masson, Paris, 1967.**[23]**J. NEČAS, "Sur la coercivité des formes sesquilinéaires elliptiques,"*Rev. Roumaine Math. Pures Appl.*, v. 9, 1964, pp. 47-69.**[24]**J. NITSCHE, " -convergence of finite element approximations,"*Mathematical Aspects of Finite Element Methods*, Rome, 1975. MR**568857 (81e:65058)****[25]**J. NITSCHE, " -convergence for finite element approximation," 2.*Conference on Finite Elements*, Rennes, France, May 1975.**[26]**J. NITSCHE, "On Dirichlet problems using subspaces with nearly zero boundary conditions,"*The Mathematical Foundations of the Finite Element Method*(A. K. Aziz, Editor), Academic Press, New York, 1972, pp. 603-627. MR**0426456 (54:14399)****[27]**J. NITSCHE & A. H. SCHATZ, "Interior estimates for Ritz-Galerkin methods,"*Math. Comp.*, v. 28, 1974, pp. 937-958. MR**0373325 (51:9525)****[28]**J. NITSCHE & A. H. SCHATZ, "On local approximation properties of projections on spline subspaces,"*Applicable Anal.*, v. 2, 1972, pp. 161-168. MR**0397268 (53:1127)****[29]**L. A. OGANESYAN & L. A. RUKHOVETS, "Variational-difference schemes for linear second-order elliptic equations in a two-dimensional region with piecewise smooth boundary,"*Ž. Vyčisl. Mat. i Mat. Fiz.*, v. 8, 1968, pp. 97-114 =*USSR Comput. Math. and Math. Phys.*, v. 8, 1968, pp. 129-152. MR**0233525 (38:1846)****[30]**T. H. H. PIAN, "Crack elements,"*Proc. World Congress on Finite Element Methods in Structural Mechanics*, Vol. I (J. Robinson, Editor), Robinson & Assoc., Verwood, Dorset, England, 1975, F.1-F.39.**[31]**A. H. SCHATZ & L. B. WAHLBIN, "Interior maximum norm estimates for finite element methods,"*Math. Comp.*, v. 31, 1977, pp. 414-442. MR**0431753 (55:4748)****[32]**R. SCOTT, "Optimal estimates for the finite element method on irregular meshes,"*Math. Comp.*, v. 30, 1976, pp. 681-697. MR**0436617 (55:9560)****[33]**L. N. SLOBODECKII, "Generalized Sobolev spaces and their application to boundary problems for partial differential equations," English transl.,*Amer. Math. Soc. Transl.*(2), v. 57, 1966, pp. 207-276.**[34]**E. M. STEIN,*Singular Integrals and Differentiability Properties of Functions*, Princeton Univ. Press, Princeton, N. J., 1970. MR**0290095 (44:7280)****[35]**G. STRANG, "Approximation in the finite element method,"*Numer. Math.*, v. 19, 1972, pp. 81-98. MR**0305547 (46:4677)****[36]**R. W. THATCHER, "The use of infinite grid refinements at singularities in the solution of Laplace's equation,"*Numer. Math.*, v. 25, 1976, pp. 163-178. MR**0400748 (53:4578)****[37]**S. V. USPENSKII, "An embedding theorem for S. L. Sobolev's classes of fractional order ,"*Dokl. Akad. Nauk SSSR*, v. 130, 1960, pp. 992-993 =*Soviet Math. Dokl.*, v. 1, 1960, pp. 132-133. MR**0124731 (23:A2042a)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65N30

Retrieve articles in all journals with MSC: 65N30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1978-0502065-1

Article copyright:
© Copyright 1978
American Mathematical Society