Approximate polynomial preconditionings applied to biharmonic equations

✍ Scribed by Yau Shu Wong; Hong Jiang

• Book ID: 104633742
• Publisher: Springer US
• Year: 1989
• Tongue: English
• Weight: 953 KB
• Volume: 3
• Category: Article
• ISSN: 0920-8542
• DOI: 10.1007/bf00129846

✦ Synopsis

Applying a finite difference approximation to a biharmonic equation results in a very ill conditioned system of equations. This paper e~mines the conjugate gradient method used with polynomial preconditioning techniques for solving such linear systems. A new approach using an approximate polynomial preconditioner is described. The preconditioner is constructed from a series approximation based on the Laplacian finite difference matrix. A particularly attractive feature of this approach is that the Laplacian matrix consists of far fewer non-zero entries than the biharmonic finite difference matrix. Moreover, analytical estimates and computational results show that this preconditioner is more effective (in terms of the rate of convergence and the computational work required per iteration) than the polynomial preconditioner based on the original biharmonic matrix operator. The conjugate gradient algorithm and the preconditioning step can be efficiently implemented on a vector supercomputer such as the CDC CYBER 205.