Efficient numerical solution of the Allen–Cahn equation via a non-lumping Galerkin finite element method

This work elaborates and investigates an effective numerical method for solving nonlinear Allen–Cahn equation. In particular, we focused on a time discretization scheme using the Crank–Nicolson (CN) method combined with space discretization using the non-lumping Galerkin finite element method. Fully discrete solutions address nonlinear terms that lose their real characteristics. Our approach avoids the lumping or linearization of the CN scheme and hence holds the accuracy regarding the treatment of non-linearity of this method. In addition, a priori error estimates concerning the semi-discrete scheme in the L∞(L2)-norm are derived. In addition, we investigated the stability of the fully discrete scheme to ensure its robustness. Numerical results are presented for problems with three different boundary conditions, and it is seen that these results are consistent with the analytical solution and exhibit good convergence with high accuracy.