Mathematics of Computation

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



Interior estimates for Ritz-Galerkin methods

Authors: Joachim A. Nitsche and Alfred H. Schatz
Journal: Math. Comp. 28 (1974), 937-958
MSC: Primary 65N30; Secondary 35JXX
MathSciNet review: 0373325
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Interior a priori error estimates in Sobolev norms are derived from interior Ritz-Galerkin equations which are common to a class of methods used in approximating solutions of second order elliptic boundary value problems. The estimates are valid for a large class of piecewise polynomial subspaces used in practice, which are defined on both uniform and nonuniform meshes. It is shown that the error in an interior domain $ \Omega $ can be estimated with the best order of accuracy that is possible locally for the subspaces used plus the error in a weaker norm over a slightly larger domain which measures the effects from outside of the domain $ \Omega $. Additional results are given in the case when the subspaces are defined on a uniform mesh. Applications to specific boundary value problems are given.

References [Enhancements On Off] (What's this?)

  • [1] Shmuel Agmon, Lectures on elliptic boundary value problems, Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. MR 0178246
  • [2] J. P. AUBIN, "Approximation des problèmes aux limites non homogènes et régularité de la convergence," Calcolo, v. 6, 1969, pp. 117--139.
  • [3] I. BABUŠKA, The Finite Element Method with Lagrangian Multipliers, Technical Note BN-724, Institute for Fluid Dynamics and Appl. Math., University of Maryland, College Park, Md., 1972.
  • [4] I. BABUŠKA, Numerical Solution of Boundary Value Problems by the Perturbed Variational Principle, Technical Note BN-624, University of Maryland, College Park, Md., 1969.
  • [5] Ju. M. Berezans′kiĭ, Expansions in eigenfunctions of selfadjoint operators, Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968. MR 0222718
  • [6] J. H. Bramble and S. R. Hilbert, Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. Math. 16 (1970/1971), 362–369. MR 0290524
  • [7] J. H. Bramble and J. E. Osborn, Rate of convergence estimates for nonselfadjoint eigenvalue approximations, Math. Comp. 27 (1973), 525–549. MR 0366029, 10.1090/S0025-5718-1973-0366029-9
  • [8] James H. Bramble and Miloš Zlámal, Triangular elements in the finite element method, Math. Comp. 24 (1970), 809–820. MR 0282540, 10.1090/S0025-5718-1970-0282540-0
  • [9] 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
  • [10] R. B. KELLOGG, Interpolation Between Subspaces of a Hilbert Space, Technical Note BN-719, Institute for Fluid Dynamics and Appl. Math., University of Maryland, College Park, Md., 1972.
  • [11] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
  • [12] J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg 36 (1971), 9–15 (German). Collection of articles dedicated to Lothar Collatz on his sixtieth birthday. MR 0341903
  • [13] 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
  • [14] J. Nitsche, Umkehrsätze für Spline-Approximationen, Compositio Math. 21 (1969), 400–416 (German, with English summary). MR 0259436
  • [15] J. Nitsche, Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens, Numer. Math. 11 (1968), 346–348 (German). MR 0233502
  • [16] Joachim A. Nitsche, Interior error estimates of projection methods, Proceedings of Equadiff III (Third Czechoslovak Conf. Differential Equations and their Applications, Brno, 1972) Purkyně Univ., Brno, 1973, pp. 235–239. Folia Fac. Sci. Natur. Univ. Purkynianae Brunensis, Ser. Monograph., Tomus 1. MR 0359361
  • [17] 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
  • [18] L. SERBIN, A Computational Investigation of Least Squares and Other Projection Methods for the Approximate Solution of Boundary Value Problems, Doctoral Thesis, Cornell University, Ithaca, N. Y., 1971.
  • [19] Approximations with special emphasis on spline functions, Proceedings of a Symposium Conducted by the Mathematics Research Center, United States Army, at the University of Wisconsin, Madison, May 5–7, 1969. Edited by I. J. Schoenberg. Publication No. 23 of the Mathematics Research Center, United States Army, The University of Wisconsin, Academic Press, New York-London, 1969. MR 0251408
  • [20] Vidar Thomée and Bertil Westergren, Elliptic difference equations and interior regularity, Numer. Math. 11 (1968), 196–210. MR 0224303
  • [21] Vidar Thomée, Discrete interior Schauder estimates for elliptic difference operators., SIAM J. Numer. Anal. 5 (1968), 626–645. MR 0238505
  • [22] Miloš Zlámal, A finite element procedure of the second order of accuracy, Numer. Math. 14 (1969/1970), 394–402. MR 0256577

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30, 35JXX

Retrieve articles in all journals with MSC: 65N30, 35JXX

Additional Information

Article copyright: © Copyright 1974 American Mathematical Society