Convergence Acceleration for Iterative Finite-Element Methods

Abstract

Sequence-to-sequence transformations are examined as a means to accelerate the convergence of iterative finite-element methods; in particular, the Shanks transform, the scalar epsilon algorithm (SEA), and the vector epsilon algorithm (VEA) are applied to dynamic relaxation (DR) solutions. One of the features of these algorithms is that they operate directly on the solution iterates and, therefore, do not require modification of the finite-element code itself. The accelerators are applied to a linear spring system and a cantilever elastica. Nonlinear sequence-to-sequence transformations are found to accelerate convergence by a factor of more than 10 and, thus, to largely overcome a major drawback of iterative finite-element methods, namely their slow rate of convergence. The VEA transformation is especially stable, is invariant with respect to coordinate transformations, and is compatible with typical kinematic constraints. Therefore, it is well suited to the acceleration of finite-element solution iterates.