Convergence Analysis of a Finite Element Approximation of Minimum Action Methods

Xiaoliang Wan, Haijun Yu, Jiayu Zhai

Abstract: In this work, we address the convergence of a finite element approximation of the minimizer of the Freidlin--Wentzell (F-W) action functional for nongradient dynamical systems perturbed by small noise. The F-W theory of large deviations is a rigorous mathematical tool to study small-noise-induced transitions in a dynamical system. The central task in the application of F-W theory of large deviations is to seek the minimizer and minimum of the F-W action functional. We discretize the F-W action functional using linear finite elements and establish the convergence of the approximation through $\Gamma$-convergence.