Preconditioning and two-level multigrid methods of arbitrary degree of approximation

Authors:
O. Axelsson and I. Gustafsson

Journal:
Math. Comp. **40** (1983), 219-242

MSC:
Primary 65N50; Secondary 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1983-0679442-3

MathSciNet review:
679442

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let *h* be a mesh parameter corresponding to a finite element mesh for an elliptic problem. We describe preconditioning methods for two-level meshes which, for most problems solved in practice, behave as methods of optimal order in both storage and computational complexity. Namely, per mesh point, these numbers are bounded above by relatively small constants for all , where is small enough to cover all but excessively fine meshes.

We note that, in practice, multigrid methods are actually solved on a finite, often even a fixed number of grid levels, in which case also these methods are not asymptotically optimal as . Numerical tests indicate that the new methods are about as fast as the best implementations of multigrid methods applied on general problems (variable coefficients, general domains and boundary conditions) for all but excessively fine meshes. Furthermore, most of the latter methods have been implemented only for difference schemes of second order of accuracy, whereas our methods are applicable to higher order approximations. We claim that our scheme could be added fairly easily to many existing finite element codes.

**[1]**O. Axelsson,*A class of iterative methods for finite element equations*, Comput. Methods Appl. Mech. Engrg.**9**(1976), no. 2, 123–127. MR**0433836**, https://doi.org/10.1016/0045-7825(76)90056-6**[2]**O. Axelsson & I. Gustafsson,*A Preconditioned Conjugate Gradient Method for Finite Element Equations, Which is Stable for Rounding Errors*, Information Processing 80 (S. H. Lavington, ed.), North-Holland, Amsterdam, 1980, pp. 723-728.**[3]**R. Bank & T. Dupont,*Analysis of a Two-Level Scheme for Solving Finite Element Equations*, Report CNA-159, Center for Numerical Analysis, The University of Texas at Austin, 1980.**[4]**Achi Brandt,*Multi-level adaptive solutions to boundary-value problems*, Math. Comp.**31**(1977), no. 138, 333–390. MR**0431719**, https://doi.org/10.1090/S0025-5718-1977-0431719-X**[5]**I. Fried, "Bounds on the extremal eigenvalues of the finite element stiffness and mass matrices and their spectral condition numbers,"*J. Sound Vibration*, v. 22, 1972, pp. 407-418.**[6]**I. Gustafsson,*Stability and Rate of Convergence of Modified Incomplete Cholesky Factorization Methods*, Thesis, Report 79.02R, Department of Computer Sciences, Chalmers University of Technology, Göteborg, Sweden, 1979.**[7]**P. W. Hemker,*Introduction to multigrid methods*, Colloquium: Numerical Solution of Partial Differential Equations (Delft, Nijmegen, Amsterdam, 1980) MC Syllabus, vol. 44, Math. Centrum, Amsterdam, 1980, pp. 59–97. MR**607518****[8]**David S. Kershaw,*The incomplete Cholesky-conjugate gradient method for the iterative solution of systems of linear equations*, J. Computational Phys.**26**(1978), no. 1, 43–65. MR**0488669****[9]**J. A. Meijerink and H. A. van der Vorst,*An iterative solution method for linear systems of which the coefficient matrix is a symmetric 𝑀-matrix*, Math. Comp.**31**(1977), no. 137, 148–162. MR**0438681**, https://doi.org/10.1090/S0025-5718-1977-0438681-4**[10]**U. Trottenberg, Private communication, 1981.

Retrieve articles in *Mathematics of Computation*
with MSC:
65N50,
65N30

Retrieve articles in all journals with MSC: 65N50, 65N30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0679442-3

Article copyright:
© Copyright 1983
American Mathematical Society