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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

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Least squares methods for $2m$th order elliptic boundary-value problems
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by J. H. Bramble and A. H. Schatz PDF
Math. Comp. 25 (1971), 1-32 Request permission

Abstract:

In this paper we consider a general class of boundary-value problems for 2mth order elliptic equations including nonhomogeneous essential boundary conditions and nonselfadjoint problems. Approximation methods involving least squares approximation of the data are presented and corresponding error estimates are proved. These methods can be considered in the category of Rayleigh-Ritz-Galerkin methods and have the special feature that the trial functions need not satisfy the boundary conditions. A special case of the trial functions which is studied are spline functions defined on a uniform mesh of width h (or more generally piecewise polynomial functions). For a given "well set" boundary-value problem for a 2mth order operator the theory presented will provide a method with any prescribed order of accuracy r which is optimal in the sense that the best approximation in the underlying subspace is of order of accuracy r.
References
  • Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246
  • Jean-Pierre Aubin, Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Gelerkin’s and finite difference methods, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 21 (1967), 599–637. MR 233068
  • Jean-Pierre Aubin, Interpolation et approximation optimales et “spline functions”, J. Math. Anal. Appl. 24 (1968), 1–24 (French). MR 231096, DOI 10.1016/0022-247X(68)90046-2
  • J. P. Aubin, "Approximation des problèms aux limites non homogènes et rĂŠgularite de la convergence," Calcolo, v. 6, 1969, pp. 117-139. J. P. Aubin, Variational Methods for Non-Homogeneous Boundary Value Problems, SYNSPADE, 1970 (B. E. Hubbard, editor). (To appear.) I. BabuĹĄka, Numerical Solution of Boundary Value Problems by the Perturbed Variational Principle, University of Maryland, Technical Note BN-624, College Park, Md., 1969. I. BabuĹĄka, Approximation by Hill Functions, University of Maryland, Technical Note BN-648, College Park, Md. I. BabuĹĄka, The Finite Element Method for Elliptic Equations With Discontinuous Coefficients, University of Maryland, Technical Note BN-631, College Park, Md. I. BabuĹĄka, Error-Bounds for Finite Element Method, University of Maryland, Technical Note BN-630, College Park, Md. I. BabuĹĄka, Computation of Derivatives in the Finite Element Method, University of Maryland, Technical Note BN-650, College Park, Md. I. BabuĹĄka, Finite Element Method for Domains With Corners, University of Maryland, Technical Note BN-636, College Park, Md. Ju. M. Berezanskii, Expansion in Eigenfunctions of Self Adjoint Operators, Naukova Dumka, Kiev, 1965; English transl., Transl. Math. Monographs, vol. 17, Amer. Math. Soc., Providence, R.I., 1968. MR 36 #5768; 5769.
  • J. H. Bramble and S. R. Hilbert, Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. Math. 16 (1970/71), 362–369. MR 290524, DOI 10.1007/BF02165007
  • James H. Bramble and Alfred H. Schatz, Rayleigh-Ritz-Galerkin methods for Dirichlet’s problem using subspaces without boundary conditions, Comm. Pure Appl. Math. 23 (1970), 653–675. MR 267788, DOI 10.1002/cpa.3160230408
  • J. H. Bramble and A. H. Schatz, On the numerical solution of elliptic boundary value problems by least squares approximation of the data, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 107–131. MR 0273837
  • James H. Bramble and MiloĹĄ ZlĂĄmal, Triangular elements in the finite element method, Math. Comp. 24 (1970), 809–820. MR 282540, DOI 10.1090/S0025-5718-1970-0282540-0
  • Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York, Inc., New York, 1967. MR 0230022, DOI 10.1007/978-3-642-46066-1
  • F. Di Guglielmo, Construction d’approximations des espaces de Sobolev sur des rĂŠseaux en simplexes, Calcolo 6 (1969), 279–331. MR 433113, DOI 10.1007/BF02576159
  • George Fix and Gilbert Strang, Fourier analysis of the finite element method in Ritz-Galerkin theory, Studies in Appl. Math. 48 (1969), 265–273. MR 258297, DOI 10.1002/sapm1969483265
  • K. O. Friedrichs and H. B. Keller, A finite difference scheme for generalized Neumann problems, Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965) Academic Press, New York, 1966, pp. 1–19. MR 0203956
  • Pierre Grisvard, CommutativitĂŠ de deux foncteurs d’interpolation et applications, J. Math. Pures Appl. (9) 45 (1966), 143–206 (French). MR 221309
  • S. Hilbert, Numerical Methods for Elliptic Boundary Problems, Ph.D. Thesis, University of Maryland, College Park, Md., 1969.
  • J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches MathĂŠmatiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
  • Enrico Magenes, Spazi d’interpolazione ed equazioni a derivate parziali, Atti del Settimo Congresso dell’Unione Matematica Italiana (Genova, 1963) Edizioni Cremonese, Rome, 1965, pp. 134–197 (Italian). MR 0215082
  • J. Nitsche, Lineare Spline-Funktionen und die Methoden von Ritz fĂźr elliptische Randwertprobleme, Arch. Rational Mech. Anal. 36 (1970), 348–355 (German). MR 255043, DOI 10.1007/BF00282271
  • J. Nitsche, Über ein Variationsprinzip zur LĂśsung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg 36 (1971), 9–15 (German). MR 341903, DOI 10.1007/BF02995904
  • Ja. A. RoÄ­tberg and Z. G. Ĺ eftel′, A homeomorphism theorem for elliptic systems, and its applications. , Mat. Sb. (N.S.) 78 (120) (1969), 446–472 (Russian). MR 0247270
  • Martin Schechter, On $L^{p}$ estimates and regularity. II, Math. Scand. 13 (1963), 47–69. MR 188616, DOI 10.7146/math.scand.a-10688
  • Martin H. Schultz, Approximation theory of multivariate spline functions in Sobolev spaces, SIAM J. Numer. Anal. 6 (1969), 570–582. MR 263218, DOI 10.1137/0706052
  • Martin H. Schultz, Rayleigh-Ritz-Galerkin methods for multidimensional problems, SIAM J. Numer. Anal. 6 (1969), 523–538. MR 263254, DOI 10.1137/0706047
  • Gilbert Strang, The finite element method and approximation theory, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 547–583. MR 0287723
  • Richard S. Varga, Hermite interpolation-type Ritz methods for two-point boundary value problems, Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965) Academic Press, New York, 1966, pp. 365–373. MR 0205475
  • MiloĹĄ ZlĂĄmal, On the finite element method, Numer. Math. 12 (1968), 394–409. MR 243753, DOI 10.1007/BF02161362
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Math. Comp. 25 (1971), 1-32
  • MSC: Primary 65N99
  • DOI: https://doi.org/10.1090/S0025-5718-1971-0295591-8
  • MathSciNet review: 0295591