Solution of Finite Element Problems by Preconditioned Conjugate Gradient and Lanczos Methods

摘要

The solution of nonlinear, transient finite element problems may be achieved by using step-by-step integration of the equations of motion combined with a Newton solution of the resulting nonlinear algebraic equations. The use of Newton type methods leads to a set of linear simultaneous algebraic equations whose solution gives the iterate. For very large problems the solution of the large set of linearized equations may be a formidable task - often consuming more than half of the computing effort when performed by a direct method based upon Gauss elimination. Accordingly, it is of considerable importance to investigate alternative methods to solve the problem. The present study presents results obtained by using a Preconditioned Conjugate Gradient Method (PCG) described in (7) and a Preconditioned Lanczos Method (PLM) described in (6) to solve a variety of numerical examples. Based upon results obtained it is evident that a significant reduction in overall effort, compared to direct solutions, may be achieved using the preconditioned methods.