A two-grid discretization scheme

for eigenvalue problems

Authors:
Jinchao Xu and Aihui Zhou

Journal:
Math. Comp. **70** (2001), 17-25

MSC (2000):
Primary 65L15, 65N15, 65N25, 65N30, 65N55

Published electronically:
August 17, 1999

MathSciNet review:
1677419

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A two-grid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations. With this new scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid, and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.

**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. MR**0450957****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. MR**1427468**, 10.1137/S0036142993247104**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. MR**962210**, 10.1090/S0025-5718-1989-0962210-8**4.**P. G. Ciarlet and J.-L. Lions (eds.),*Handbook of numerical analysis. Vol. II*, Handbook of Numerical Analysis, II, North-Holland, Amsterdam, 1991. Finite element methods. Part 1. MR**1115235****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. MR**716134****6.**P. G. Ciarlet and J.-L. Lions (eds.),*Handbook of numerical analysis. Vol. II*, Handbook of Numerical Analysis, II, North-Holland, Amsterdam, 1991. Finite element methods. Part 1. MR**1115235****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. MR**1312392**, 10.1090/conm/180/01971**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 (electronic). MR**1618822**, 10.1137/S0036142995293493**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. MR**1326676**, 10.1016/0096-3003(94)00134-P**10.**Qun Lin,*Some problems concerning approximate solutions of operator equations*, Acta Math. Sinica**22**(1979), no. 2, 219–230 (Chinese, with English summary). MR**542459****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. MR**1342288**, 10.1137/0732054**12.**T. Utnes,*Two-grid finite element formulations of the incompressible Navier-Stokes equations*, Comm. Numer. Methods Engrg.**13**(1997), no. 8, 675–684. MR**1466044**, 10.1002/(SICI)1099-0887(199708)13:8<675::AID-CNM98>3.0.CO;2-N**13.**Jinchao Xu,*A new class of iterative methods for nonselfadjoint or indefinite problems*, SIAM J. Numer. Anal.**29**(1992), no. 2, 303–319. MR**1154268**, 10.1137/0729020**14.**Jinchao Xu,*Iterative methods by space decomposition and subspace correction*, SIAM Rev.**34**(1992), no. 4, 581–613. MR**1193013**, 10.1137/1034116**15.**Jinchao Xu,*A novel two-grid method for semilinear elliptic equations*, SIAM J. Sci. Comput.**15**(1994), no. 1, 231–237. MR**1257166**, 10.1137/0915016**16.**Jinchao Xu,*Two-grid discretization techniques for linear and nonlinear PDEs*, SIAM J. Numer. Anal.**33**(1996), no. 5, 1759–1777. MR**1411848**, 10.1137/S0036142992232949**17.**Xu, J. and Zhou, A.(1998): Local and parallel finite element algorithms based on two-grid discretizations, Math. Comp.(to appear).

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Additional Information

**Jinchao Xu**

Affiliation:
Center for Computational Mathematics and Applications, Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802

Email:
xu@math.psu.edu

**Aihui Zhou**

Affiliation:
Institute of Systems Science, Academia Sinica, Beijing 100080, China

Email:
azhou@bamboo.iss.ac.cn

DOI:
http://dx.doi.org/10.1090/S0025-5718-99-01180-1

Keywords:
Eigenvalue problems,
finite elements,
partial differential equations,
integral equations,
two-grid method

Received by editor(s):
December 16, 1998

Received by editor(s) in revised form:
February 25, 1999

Published electronically:
August 17, 1999

Additional Notes:
This work was partially supported by NSF DMS-9706949, NSF ACI-9800244 and NASA NAG2-1236 through Penn State, and the Center for Computational Mathematics and Applications, The Pennsylvania State University, and by NSF ASC 9720257 through UCLA. The second author was also partially supported by National Science Foundation of China.

Article copyright:
© Copyright 1999
American Mathematical Society