The analysis of an efficient numerical scheme for the Allen–Cahn equations using the Galerkin method

In this paper, we propose an efficient numerical scheme for the Allen–Cahn equations. We show theoretically using the Galerkin method and the compactness theorem that the solution of the afore-mentioned equation exists and is unique in appropriate spaces with the interaction length parameter α well controlled. We further, show numerically that the proposed scheme is stable and converge optimally in the L² as well as the H¹-norms with its numerical solution preserving all the qualitative properties of the exact solution. With the help of an example and a carefully chosen α, we use numerical experiments to justify the validity of the proposed scheme.