Conservative EQ1rot nonconforming FEM for nonlinear Schrödinger equation with wave operator

Abstract

AbstractIn this paper, we consider leap‐frog finite element methods with element for the nonlinear Schrödinger equation with wave operator. We propose that both the continuous and discrete systems can keep mass and energy conservation. In addition, we focus on the unconditional superconvergence analysis of the numerical scheme, the key of which is the time‐space error splitting technique. The spatial error is derived independently with order in ‐norm, where and denote the space and time step size. Then the unconditional optimal error and superclose result with order are deduced, and the unconditional optimal error is obtained with order by using interpolation theory. The final unconditional superconvergence result with order is derived by the interpolation postprocessing technique. Furthermore, we apply the proposed leap‐frog finite element methods to solve the logarithmic Schrödinger equation with wave operator by introducing a regularized system with a small regularization parameter . The detailed theoretical conclusions, including the mass and energy conservation laws of the continuous regularized and discrete regularized systems and the convergence and superconvergence results, are presented as prolongation of the previous work. At last, some numerical experiments are given to confirm our theoretical analysis.