TY - JOUR

T1 - Simplified heuristic Fourier analysis of iteration convergence rates

AU - Ramshaw, J. D.

AU - Mesina, G. L.

PY - 1994/6

Y1 - 1994/6

N2 - Chan and Elman (1989) have recently shown that heuristic Fourier analysis can be used to determine approximate convergence rates of iterative methods. Here we present a simple procedure and useful relations that facilitate the practical application of this method. The procedure is based on the fact that the mesh spacing h is typically small, so that attention may be restricted to the asymptotic behaviour of the convergence rate R for small h. We accordingly derive simple expressions for the first few coefficients in a Taylor series expansion of R in powers of h. These coefficients are expressed in terms of derivatives with respect to h of the complex Fourier growth factor ξ rather than its modulus |ξ|. The required derivatives can then be obtained by implicit differentiation without explicitly solving for ξ, which further simplifies the analysis. The overall procedure is illustrated by applying it to the Jacobi, Gauss‐Seidel and successive overrelaxation methods, for which we obtain results asymptotically equivalent to those of Chan and Elman. We also analyse the effect of second‐order Richardson acceleration on the Jacobi method, for which we obtain an asymptotic convergence rate in precise agreement with classical analysis.

AB - Chan and Elman (1989) have recently shown that heuristic Fourier analysis can be used to determine approximate convergence rates of iterative methods. Here we present a simple procedure and useful relations that facilitate the practical application of this method. The procedure is based on the fact that the mesh spacing h is typically small, so that attention may be restricted to the asymptotic behaviour of the convergence rate R for small h. We accordingly derive simple expressions for the first few coefficients in a Taylor series expansion of R in powers of h. These coefficients are expressed in terms of derivatives with respect to h of the complex Fourier growth factor ξ rather than its modulus |ξ|. The required derivatives can then be obtained by implicit differentiation without explicitly solving for ξ, which further simplifies the analysis. The overall procedure is illustrated by applying it to the Jacobi, Gauss‐Seidel and successive overrelaxation methods, for which we obtain results asymptotically equivalent to those of Chan and Elman. We also analyse the effect of second‐order Richardson acceleration on the Jacobi method, for which we obtain an asymptotic convergence rate in precise agreement with classical analysis.

UR - http://www.scopus.com/inward/record.url?scp=0028448047&partnerID=8YFLogxK

U2 - 10.1002/cnm.1640100607

DO - 10.1002/cnm.1640100607

M3 - Article

AN - SCOPUS:0028448047

SN - 1069-8299

VL - 10

SP - 481

EP - 487

JO - Communications in Numerical Methods in Engineering

JF - Communications in Numerical Methods in Engineering

IS - 6

ER -