Analysis of the Jacobian-free multiscale method (JFMM)

Abstract

In this paper we perform in-depth analysis of Jacobian-free multiscale method (JFMM) in which explicit computation of the Jacobian matrix at the macroscale is circumvented using a Newton---Krylov process. Not having to explicitly compute and store the Jacobian matrix at each Newton step reduces storage requirements and computational costs compared to previous efforts based on homogenized material coefficients with Jacobian computation at every Newton step. We present an estimate of the optimal perturbation step-size that minimizes the finite difference approximation error associated with the Jacobian-vector product in the Jacobian-free approach. Two- and three-dimensional numerical examples demonstrate that while the rate of convergence of Newton iterations for the JFMM and the computational homogenization-based two-level finite element $$(\hbox {FE}^{2})$$(FE2) multiscale method is comparable, the computational cost of JFMM varies linearly with increasing number of degrees of freedom (n) at the macroscale, and not exponentially as in the $$\hbox {FE}^{2}$$FE2 method. The storage requirement for the method increases linearly with increasing n at the macroscale, whereas, it increases as approximately $$O(n^{8/5})$$O(n8/5) and $$O(n^{9/5})$$O(n9/5) for the $$\hbox {FE}^{2}$$FE2 method in two- and three-dimensions, respectively.