Rate of convergence estimates for nonselfadjoint eigenvalue approximations

Bramble, J. H.; Osborn, J. E.

doi:10.1090/S0025-5718-1973-0366029-9

Rate of convergence estimates for nonselfadjoint eigenvalue approximations

Authors: J. H. Bramble and J. E. Osborn
Journal: Math. Comp. 27 (1973), 525-549
MSC: Primary 65J05
DOI: https://doi.org/10.1090/S0025-5718-1973-0366029-9
MathSciNet review: 0366029
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Abstract: In this paper, a general approximation theory for the eigenvalues and corresponding subspaces of generalized eigenfunctions of a certain class of compact operators is developed. This theory is then used to obtain rate of convergence estimates for the errors which arise when the eigenvalues of nonselfadjoint elliptic partial differential operators are approximated by Rayleigh-Ritz-Galerkin type methods using finite-dimensional spaces of trial functions, e.g. spline functions. The approximation methods include several in which the functions in the space of trial functions are not required to satisfy any boundary conditions.

[1] N. Aronszajn, "The Rayleigh-Ritz and A. Weinstein methods for approximation of eigenvalues. I, II," Proc. Nat. Acad. Sci. U.S.A., v. 34, 1948, pp. 474-480, 594-601. MR 10, 382.
[2] N. Aronszajn, Approximation methods for eigenvalues of completely continuous symmetric operators, Proceedings of the Symposium on Spectral Theory and Differential Problems, Oklahoma Agricultural and Mechanical College, Stillwater, Okla., 1951, pp. 179–202. MR 0044736
[3] Nathan Aronszajn and Alexander Weinstein, Existence, convergence and equivalence in the unified theory of eigenvalues of plates and membranes, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 188–191. MR 4679, https://doi.org/10.1073/pnas.27.3.188
[4] Approximation by hill functions, Comment. Math. Univ. Carolinae 11 (1970), 787–811. MR 292309
[5] I. Babuška, Approximation by Hill Functions II, Technical Note BN-708, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Md., 1971.
[6] I. Babuška, The Finite Element Method with Lagrangian Multipliers, Technical Note BN-724, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Md., 1972.
[7] Norman W. Bazley, Lower bounds for eigenvalues, J. Math. Mech. 10 (1961), 289–307. MR 0128612
[8] Norman W. Bazley and David W. Fox, Truncations in the method of intermediate problems for lower bounds to eigenvalues, J. Res. Nat. Bur. Standards Sect. B 65B (1961), 105–111. MR 0142897
[9] Norman W. Bazley and David W. Fox, A procedure for estimating eigenvalues, J. Mathematical Phys. 3 (1962), 469–471. MR 144454, https://doi.org/10.1063/1.1724246
[10] Ju. M. Berezanskiĭ, 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; 36 #5769.
[11] Garrett Birkhoff, C. de Boor, B. Swartz, and B. Wendroff, Rayleigh-Ritz approximation by piecewise cubic polynomials, SIAM J. Numer. Anal. 3 (1966), 188–203. MR 203926, https://doi.org/10.1137/0703015
[12] James H. Bramble, Todd Dupont, and Vidar Thomée, Projection methods for Dirichlet’s problem in approximating polygonal domains with boundary-value corrections, Math. Comp. 26 (1972), 869–879. MR 343657, https://doi.org/10.1090/S0025-5718-1972-0343657-7
[13] J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 112–124. MR 263214, https://doi.org/10.1137/0707006
[14] 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, https://doi.org/10.1007/BF02165007
[15] 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, https://doi.org/10.1002/cpa.3160230408
[16] J. H. Bramble and A. H. Schatz, Least squares methods for 2𝑚th order elliptic boundary-value problems, Math. Comp. 25 (1971), 1–32. MR 295591, https://doi.org/10.1090/S0025-5718-1971-0295591-8
[17] James H. Bramble and Miloš Zlámal, Triangular elements in the finite element method, Math. Comp. 24 (1970), 809–820. MR 282540, https://doi.org/10.1090/S0025-5718-1970-0282540-0
[18] P. G. Ciarlet and P.-A. Raviart, Interpolation theory over curved elements, with applications to finite element methods, Comput. Methods Appl. Mech. Engrg. 1 (1972), 217–249. MR 375801, https://doi.org/10.1016/0045-7825(72)90006-0
[19] P. G. Ciarlet, M. H. Schultz, and R. S. Varga, Numerical methods of high-order accuracy for nonlinear boundary value problems. III. Eigenvalue problems, Numer. Math. 12 (1968), 120–133. MR 233517, https://doi.org/10.1007/BF02173406
[20] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. MR 0188745
[21] Gaetano Fichera, Approximation and estimates for eigenvalues, Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965) Academic Press, New York, 1966, pp. 317–352. MR 0217644
[22] Gaetano Fichera, Further developments in the approximation theory of eigenvalues, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 243–252. MR 0277104
[23] F. Di Guglielmo, Construction d’approximations des espaces de Sobolev sur des réseaux en simplexes, Calcolo 6 (1969), 279–331. MR 433113, https://doi.org/10.1007/BF02576159
[24] S. Hilbert, Numerical Methods for Elliptic Boundary Problems, Thesis, University of Maryland, College Park, Md., 1969.
[25] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
[26] 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
[27] I. Marek, Approximation of the Principal Eigenelements in K-Positive Non Self-Adjoint Eigenvalue Problems, MRC Technical Summary Report #1094, University of Wisconsin, Madison, Wis., 1971.
[28] 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). MR 341903, https://doi.org/10.1007/BF02995904
[29] J. Nitsche, "A projection method for Dirichlet-problems using subspaces with nearly zero boundary conditions." (Preprint.)
[30] John E. Osborn, Approximation of the eigenvalues of non self-adjoint operators, J. Math. and Phys. 45 (1966), 391–401. MR 208379
[31] John E. Osborn, Approximation of the eigenvalues of a class of unbounded, nonself-adjoint operators, SIAM J. Numer. Anal. 4 (1967), 45–54. MR 213904, https://doi.org/10.1137/0704005
[32] J. E. Osborn, "A method for approximating the eigenvalues of non self-adjoint ordinary differential operators," Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. (4), no. 14. pp. 1-56.
[33] J. G. Pierce and R. S. Varga, Higher order convergence results for the Rayleigh-Ritz method applied to eigenvalue problems. I. Estimates relating Rayleigh-Ritz and Galerkin approximations to eigenfunctions, SIAM J. Numer. Anal. 9 (1972), 137–151. MR 395268, https://doi.org/10.1137/0709014
[34] J. G. Pierce and R. S. Varga, Higher order convergence results for the Rayleigh-Ritz method applied to eigenvalue problems. II. Improved error bounds for eigenfunctions, Numer. Math. 19 (1972), 155–169. MR 323133, https://doi.org/10.1007/BF01402526
[35] Martin Schechter, On 𝐿^{𝑝} estimates and regularity. I, Amer. J. Math. 85 (1963), 1–13. MR 188615, https://doi.org/10.2307/2373179
[36] Martin Schechter, On 𝐿^{𝑝} estimates and regularity. II, Math. Scand. 13 (1963), 47–69. MR 188616, https://doi.org/10.7146/math.scand.a-10688
[37] I. J. Schoenberg, "Contributions to the problem of approximation of equidistant data by analytic functions," Quart. Appl. Math., v. 4, 1946, part A, pp. 45-99, part B, pp. 112-141. MR 7, 487; 8, 55.
[38] Martin H. Schultz, Rayleigh-Ritz-Galerkin methods for multidimensional problems, SIAM J. Numer. Anal. 6 (1969), 523–538. MR 263254, https://doi.org/10.1137/0706047
[39] Martin H. Schultz, Multivariate spline functions and elliptic problems, Approximations with Special Emphasis on Spline Functions (Proc. Sympos. Univ. of Wisconsin, Madison, Wis., 1969) Academic Press, New York, 1969, pp. 279–347. MR 0257560
[40] Martin H. Schultz, 𝐿² error bounds for the Rayleigh-Ritz-Galerkin method, SIAM J. Numer. Anal. 8 (1971), 737–748. MR 298918, https://doi.org/10.1137/0708067
[41] William Stenger, On the variational principles for eigenvalues for a class of unbounded operators, J. Math. Mech. 17 (1967/1968), 641–648. MR 0227800
[42] G. M. Vainikko, Asymptotic error bounds for projection methods in the eigenvalue problem, Ž. Vyčisl. Mat i Mat. Fiz. 4 (1964), 405–425 (Russian). MR 176340
[43] G. M. Vainikko, On the rate of convergence of certain approximation methods of Galerkin type in eigenvalue problems, Izv. Vysš. Učebn. Zaved. Matematika 1966 (1966), no. 2 (51), 37–45 (Russian). MR 0198669
[44] G. M. Vainikko, "On the speed of convergence of approximate methods in the eigenvalue problem," Ž. Vyčisl. Mat. i Mat. Fiz., v. 7, 1967, pp. 977-987. USSR Comput. Math. and Math. Phys., v. 7, 1967, pp. 18-32.
[45] H. F. Weinberger, Error estimation in the Weinstein method for eigenvalues, Proc. Amer. Math. Soc. 3 (1952), 643–646. MR 50177, https://doi.org/10.1090/S0002-9939-1952-0050177-5
[46] H. F. Weinberger, A Theory of Lower Bounds for Eigenvalues, Technical Note BN-183, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Md., 1959.
[47] A. Weinstein, "Sur la stabilité des plaques encastrées," C. R. Acad. Sci. Paris, v. 200, 1935, pp. 107-109.
[48] A. Weinstein, "Étude des spectres des équations aux dérivées partielles de la théorie des plaques élastiques," Mem. Sci. Math., v. 88, 1937.
[49] Alexander Weinstein, Bounds for eigenvalues and the method of intermediate problems, Partial differential equations and continuum mechanics, Univ. of Wisconsin Press, Madison, Wis., 1961, pp. 39–53. MR 0126068
[50] A. Weinstein, A necessary and sufficient condition in the maximum-minimum theory of eigenvalues, Studies in mathematical analysis and related topics, Stanford Univ. Press, Stanford, Calif., 1962, pp. 429–434. MR 0149657
[51] Alexander Weinstein, The intermediate problems and the maximum-minimum theory of eigenvalues, J. Math. Mech. 12 (1963), 235–245. MR 0155083
[52] Alexander Weinstein, An invariant fomulation of the new maximum-minimum theory of eigenvalues, J. Math. Mech. 16 (1966), 213–218. MR 0212604, https://doi.org/10.1512/iumj.1967.16.16015
[53] O. C. Zienkiewicz, The Finite Element Method in Structural and Continuum Mechanics, McGraw-Hill, New York, 1967.
[54] Miloš Zlámal, On the finite element method, Numer. Math. 12 (1968), 394–409. MR 243753, https://doi.org/10.1007/BF02161362
[55] M. Zlámal, "Curved elements in the finite element method." (Preprint.)