Quasi-newtonian methods for the solution of hierarchical finite element equations

Abstract

This paper exploits the solution of hierarchical finite element equations via quasi-Newtonian (QN) methods, and discovers that QN methods can compete with the preconditioned conjugate gradient (PCG) method in the rate of convergence. The currently available version of the PCG method is unable to solve the indefinite system of equations, because the preconditioning matrices involve the arithmetic operations on the square roots of diagonal pivots of the coefficient matrix. On the contrary, the QN methods investigated in this paper do not have this limitation. Thus, they have the potential to become popular solution strategies for nonlinear hierarchical finite element analysis in the future. Details on how the QN algorithms are implemented for solving the linear hierarchical system of equations are presented. The QN algorithms as well as the PCG method are tested by using an example in which the coefficient matrices are symmetric positive definite. Comparisons on computational efficiency and convergence rate for the positive definite case are then discussed. Also, the capability of QN algorithms for solving the indefinite system of equations is also presented via a numerical experiment using another example.