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 Free Access

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]**Ivo Babuška,*Finite element method for domains with corners*, Computing (Arch. Elektron. Rechnen)**6**(1970), 264–273 (English, with German summary). MR**0293858****[2]**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****[3]**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**0455462**, https://doi.org/10.1137/0713021**[4]**I. BABUŠKA & W. RHEINBOLDT, "Mathematical problems of computational decisions in the finite element method," Technical Note TR-426, University of Maryland, 1975.**[5]**Ivo Babuška and Michael B. Rosenzweig,*A finite element scheme for domains with corners*, Numer. Math.**20**(1972/73), 1–21. MR**0323129**, https://doi.org/10.1007/BF01436639**[6]**James H. Bramble and Miloš Zlámal,*Triangular elements in the finite element method*, Math. Comp.**24**(1970), 809–820. MR**0282540**, https://doi.org/10.1090/S0025-5718-1970-0282540-0**[7]**Philippe G. Ciarlet,*Numerical analysis of the finite element method*, Les Presses de l’Université de Montréal, Montreal, Que., 1976. Séminaire de Mathématiques Supérieures, No. 59 (Été 1975). MR**0495010****[8]**P. G. Ciarlet and P.-A. Raviart,*General Lagrange and Hermite interpolation in 𝑅ⁿ with applications to finite element methods*, Arch. Rational Mech. Anal.**46**(1972), 177–199. MR**0336957**, https://doi.org/10.1007/BF00252458**[9]**Ph. Clément,*Approximation by finite element functions using local regularization*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. \jname RAIRO Analyse Numérique**9**(1975), no. R-2, 77–84 (English, with Loose French summary). MR**0400739****[10]**Jim Douglas Jr., Todd Dupont, and Mary Fanett Wheeler,*An 𝐿^{∞} 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****[11]**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****[12]**J. Frehse and R. Rannacher,*Eine 𝐿¹-Fehlerabschätzung für diskrete Grundlösungen in der Methode der finiten Elemente*, Finite Elemente (Tagung, Univ. Bonn, Bonn, 1975) Inst. Angew. Math., Univ. Bonn, Bonn, 1976, pp. 92–114. Bonn. Math. Schrift., No. 89 (German, with English summary). MR**0471370****[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]**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****[15]**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**0331715**, https://doi.org/10.1090/S0025-5718-1973-0331715-3**[16]**Pierre Jamet,*Estimations d’erreur pour des éléments finis droits presque dégénérés*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. \jname RAIRO Analyse Numérique**10**(1976), no. R-1, 43–60 (French, with Loose English summary). MR**0455282****[17]**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****[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 value problems for elliptic equations in domains with conical or angular points*, Trudy Moskov. Mat. Obšč.**16**(1967), 209–292 (Russian). MR**0226187****[20]**J. L. LIONS & E. MAGENES,*Problèmes aux Limites Non Homogènes et Applications*, I, Dunod, Paris, 1968.**[21]**Frank Natterer,*Über die punktweise Konvergenz finiter Elemente*, Numer. Math.**25**(1975/76), no. 1, 67–77 (German, with English summary). MR**0474884**, https://doi.org/10.1007/BF01419529**[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. A. Nitsche,*𝐿_{∞}-convergence of finite element approximation*, Journées “Éléments Finis”}, address=Rennes, date=1975, (1975)**[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 with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 603–627. MR**0426456****[27]**Joachim A. Nitsche and Alfred H. Schatz,*Interior estimates for Ritz-Galerkin methods*, Math. Comp.**28**(1974), 937–958. MR**0373325**, https://doi.org/10.1090/S0025-5718-1974-0373325-9**[28]**J. Nitsche and A. Schatz,*On local approximation properties of 𝐿₂-projection on spline-subspaces*, Applicable Anal.**2**(1972), 161–168. Collection of articles dedicated to Wolfgang Haack on the occasion of his 70th birthday. MR**0397268**, https://doi.org/10.1080/00036817208839035**[29]**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**0233525****[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 and L. B. Wahlbin,*Interior maximum norm estimates for finite element methods*, Math. Comp.**31**(1977), no. 138, 414–442. MR**0431753**, https://doi.org/10.1090/S0025-5718-1977-0431753-X**[32]**Ridgway Scott,*Optimal 𝐿^{∞} estimates for the finite element method on irregular meshes*, Math. Comp.**30**(1976), no. 136, 681–697. MR**0436617**, https://doi.org/10.1090/S0025-5718-1976-0436617-2**[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]**Elias M. Stein,*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095****[35]**Gilbert Strang,*Approximation in the finite element method*, Numer. Math.**19**(1972), 81–98. MR**0305547**, https://doi.org/10.1007/BF01395933**[36]**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**0400748**, https://doi.org/10.1007/BF01462270**[37]**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**

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