Application of a Modified Algorithm of the Method of Conjugate Gradients in Finite-Element Analyses

Abstract

For the solution of systems of linear algebraic equations by the finite-element method, we consider a generalized method of conjugate gradients with preconditioning matrix constructed by using the transition matrix for the method of symmetric upper relaxation. It is shown that the rate of the iterative process can be doubled. We present the results of the numerical analysis of the rate of convergence of the iterative process in the solution of two-dimensional model problems of the theory of elasticity and linear fracture mechanics with the help of the classical and modified algorithms of the method of conjugate gradients with preconditioning matrix of the method of symmetric upper relaxation.