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In this work, we address the convergence of a finite element approximation ofthe minimizer of the Freidlin-Wentzell (F-W) action functional for non-gradientdynamical systems perturbed by small noise. The F-W theory of large deviationsis a rigorous mathematical tool to study small-noise-induced transitions in adynamical system. The central task in the application of F-W theory of largedeviations is to seek the minimizer and minimum of the F-W action functional.We discretize the F-W action functional using linear finite elements, andestablish the convergence of {the approximation} through $\Gamma$-convergence.

Convergence analysis of a finite element approximation of minimum action methods