Analysis of a second order discontinuous Galerkin finite element method for the Allen-Cahn equation
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The paper proposes and analyzes an efficient second-order in time numerical approximation for the Allen-Cahn equation which is a nonlinear singular perturbation of the reaction-diffusion model arising from phase separation in alloys. We firstly present a fully discrete, nonlinear interior penalty discontinuous Galerkin finite element(IPDGFE) method, which is based on the modified Crank-Nicolson scheme and a mid-point approximation of the potential term f(u). We then derive the stability analysis and error estimates for the proposed IPDGFE method under some regularity assumptions on the initial function u0. There are two key works in our analysis, one is to establish unconditionally energy-stable scheme for the discrete solutions. The other is to use a discrete spectrum estimate to handle the midpoint of the discrete solutions um and um+1 in the nonlinear term, instead of using the standard Gronwall inequality technique. We obtain that all our error bounds depend on reciprocal of the perturbation parameter ε only in some lower polynomial order, instead of exponential order.