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In this work, we propose a bi-grid scheme framework for the Allen-Cahnequation in Finite Element Method. The new methods are based on the use of twoFEM spaces, a coarse one and a fine one, and on a decomposition of the solutioninto mean and fluctuant parts. This separation of the scales, in both space andfrequency, allows to build a stabilization on the high modes components: themain computational effort is concentrated on the coarse space on which animplicit scheme is used while the fluctuant components of the fine space areupdated with a simple semi-implicit scheme, they are smoothed without damagingthe consistency. The numerical examples we give show the good stability and therobustness of the new methods. An important reduction of the computation timeis also obtained when comparing our methods with fully implicit ones.

A Stabilized bi-grid method for Allen Cahn equation in Finite Elements