Unconditional superconvergence analysis of a linearized Galerkin FEM for nonlinear Schrödinger equation

Abstract

A linearized backward Euler scheme with conforming finite element method for nonlinear Schödinger equation is studied, and the unconditional superconvergence result is deduced. A time‐discrete system is introduced to split the error into 2 parts, the temporal error and the spatial error. Firstly, a rigorous analysis for the regularity of the time‐discrete system is presented based on the proof of the temporal error skillfully. Secondly, the classical Ritz projection is applied to get the spatial error with order O(h2) in L2‐norm, which plays an important role in getting rid of the restriction of τ. Then, superclose estimates of order O(h2 + τ) in H1‐norm is arrived at with the above achievements and the relationship between the classical Ritz projection and the interpolated operator. Thirdly, global superconvergence result is deduced through interpolated postprocessing technique. At last, a numerical example is provided to confirm the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step.