A two-grid discretization scheme for eigenvalue problems. (English) Zbl 0959.65119

Xu, Jinchao; Zhou, Aihui

Math. Comput. 70, No. 233, 17-25 (2001).

A two-grid discretization technique is presented for solving linear eigenvalue problems posed in variational form, \[ a(u,v)= \lambda b(u,v)\quad\text{for all }v\in X. \] Here, \(X\) denotes a real Hilbert space, and \(a( ,)\) and \(b( ,)\) are symmetric and positive bilinear forms such that the problem exhibits a point spectrum \(0<\lambda_1\leq \lambda_2\leq\cdots\). Two linear finite element subspaces \(S^H\subset S^h\subset X\) are introduced which are defined on a coarse grid \(T^H\) and a refined grid \(T^h\), respectively. Given on the coarse grid an approximate eigensolution \(u^H\in S^H\), \(\lambda^H\in R^1\), then \(u^h\in S^h\), \(\lambda\in R^1\) is defined by \(a(u^h, v)= \lambda^Hb(u^H, v)\) for all \(v\in s^h\) together with \(\lambda^h= a(u^h, u^h)/b(u^h, u^h)\). Asymptotic error estimates in terms of \(H\) and \(h\) are presented. The technique is illustrated by simple numerical experiments for the first Laplace eigenvalue problem on the unit square.

Reviewer: Waldemar Velte (Veitshöchheim)

MSC:

65N25	Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N55	Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35P15	Estimates of eigenvalues in context of PDEs

Keywords:

finite elements; two-grid method; linear eigenvalue problems; Hilbert space; finite element; error estimates; numerical experiments; first Laplace eigenvalue problem

References:

[1]	Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. · Zbl 0314.46030
[2]	O. Axelsson and W. Layton, A two-level discretization of nonlinear boundary value problems, SIAM J. Numer. Anal. 33 (1996), no. 6, 2359 – 2374. · Zbl 0866.65077
[3]	I. Babuška and J. E. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comp. 52 (1989), no. 186, 275 – 297. · Zbl 0675.65108
[4]	P. G. Ciarlet and J.-L. Lions , Handbook of numerical analysis. Vol. II, Handbook of Numerical Analysis, II, North-Holland, Amsterdam, 1991. Finite element methods. Part 1. · Zbl 0712.65091
[5]	Françoise Chatelin, Spectral approximation of linear operators, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With a foreword by P. Henrici; With solutions to exercises by Mario Ahués.
[6]	P. G. Ciarlet and J.-L. Lions , Handbook of numerical analysis. Vol. II, Handbook of Numerical Analysis, II, North-Holland, Amsterdam, 1991. Finite element methods. Part 1. · Zbl 0712.65091
[7]	Clint N. Dawson and Mary F. Wheeler, Two-grid methods for mixed finite element approximations of nonlinear parabolic equations, Domain decomposition methods in scientific and engineering computing (University Park, PA, 1993) Contemp. Math., vol. 180, Amer. Math. Soc., Providence, RI, 1994, pp. 191 – 203. · Zbl 0817.65080
[8]	Clint N. Dawson, Mary F. Wheeler, and Carol S. Woodward, A two-grid finite difference scheme for nonlinear parabolic equations, SIAM J. Numer. Anal. 35 (1998), no. 2, 435 – 452. · Zbl 0927.65107
[9]	W. Layton and W. Lenferink, Two-level Picard and modified Picard methods for the Navier-Stokes equations, Appl. Math. Comput. 69 (1995), no. 2-3, 263 – 274. · Zbl 0828.76017
[10]	Qun Lin, Some problems concerning approximate solutions of operator equations, Acta Math. Sinica 22 (1979), no. 2, 219 – 230 (Chinese, with English summary). · Zbl 0397.65070
[11]	Martine Marion and Jinchao Xu, Error estimates on a new nonlinear Galerkin method based on two-grid finite elements, SIAM J. Numer. Anal. 32 (1995), no. 4, 1170 – 1184. · Zbl 0853.65092
[12]	T. Utnes, Two-grid finite element formulations of the incompressible Navier-Stokes equations, Comm. Numer. Methods Engrg. 13 (1997), no. 8, 675 – 684. , https://doi.org/10.1002/(SICI)1099-0887(199708)13:83.0.CO;2-N · Zbl 0883.76052
[13]	Jinchao Xu, A new class of iterative methods for nonselfadjoint or indefinite problems, SIAM J. Numer. Anal. 29 (1992), no. 2, 303 – 319. · Zbl 0756.65050
[14]	Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581 – 613. · Zbl 0788.65037
[15]	Jinchao Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput. 15 (1994), no. 1, 231 – 237. · Zbl 0795.65077
[16]	Jinchao Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33 (1996), no. 5, 1759 – 1777. · Zbl 0860.65119
[17]	Xu, J. and Zhou, A.(1998): Local and parallel finite element algorithms based on two-grid discretizations, Math. Comp.(to appear).