Fourier analysis of iterative schemes for solving the biharmonic equation

Abstract

Fourier analysis is applied to study the rate of convergence of stationary iterative algorithms for solving 13-point finite difference scheme arising from the discretization of the biharmonic equation. The number of iterations required for convergence of the conventional point Gauss Seidel method is shown to be of order O(h ⁻⁴) while the conventional successive-over-relaxation (SOR) scheme with optimal relaxation parameter improves the order to O(h ⁻³).Here, h is the mesh size. We employ Fourier analysis to derive new Gauss-Seidel and SOR schemes whose number of iterations required for convergence are of order O(h ⁻²) and O(h ⁻¹) respectively. The method of Fourier analysis is found to be a more direct and effective tool for deriving efficient iterative schemes compared to Garabedian's method of differential eigenvalue problem.

Keywords:

• Fourier analysis
• iterative schemes
• biharmonic equation

C.R. Categories:

• G.1.8