An efficient nonconforming finite element two-grid method for Allen–Cahn equation

Abstract

In this paper, superconvergent analysis of an efficient two-grid method is discussed for the Allen–Cahn equation with the nonconforming EQ1rot finite element. The unconditional stability of the numerical scheme is proved based on the monotonically increasing character of the nonlinear term in the problem, while the previous works always require some certain stability conditions. By use of the typical properties of EQ1rot element together with a more accurate estimate on the nonlinear term, the superclose result of order O(h2+H4+τ) in the broken H1-norm is deduced rigorously without the above restrictions for the first time. Furthermore, the global superconvergence behavior is derived through interpolated postprocessing skill. Numerical results illustrate that the proposed method is actually effective.