Convergence rate estimates for iterative solutions of the biharmonic equation. (English) Zbl 0675.65104
The article is devoted to an analysis of the spectral radius of iteration matrices of finite-difference schemes approximating the biharmonic equation. The authors use the idea of P. R. Garabedian [Math. Tables Aids Comput. 10, 183-185 (1956; Zbl 0073.108)] according to which the estimation of this radius in the form \(\rho =1-O(h^ k)\) can be reduced to the eigenvalue problem for some partial-differential operator. It is shown that the obtained estimates are precise for many classical iteration schemes. They can give also information on how much successful overrelaxation can improve the convergence rate. Some new iterative technics are suggested for which \(\rho =1-O(h)\).
Reviewer: V.Korneev
MSC:
65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
65F10 | Iterative numerical methods for linear systems |
35J40 | Boundary value problems for higher-order elliptic equations |
31A30 | Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions |
Keywords:
spectral radius; iteration matrices; finite-difference schemes; biharmonic equation; overrelaxation; convergence rateCitations:
Zbl 0073.108References:
[1] | Garabedian, P. R., Estimation of the relaxation factor for small mesh size, Mathematical Tables and other Aids to Computation, 10, 183-185 (1956) · Zbl 0073.10804 |
[2] | Varga, R. S., Matrix Iterative Analysis (1962), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0133.08602 |
[3] | D. Young, Iterative Solution of Large Linear Systems; D. Young, Iterative Solution of Large Linear Systems · Zbl 0231.65034 |
[4] | Hagemann, L.; Young, D., Applied Iterative Methods (1981), Academic Press: Academic Press New York · Zbl 0459.65014 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.