Conservative EQ1rot nonconforming FEM for nonlinear Schrödinger equation with wave operator

Abstract

In this paper, we consider leap‐frog finite element methods with EQ1rot$$ {\mathrm{EQ}}_1^{\mathrm{rot}} $$ 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 τ$$ \tau $$ independently with order O(h2+hτ)$$ O\left({h}^2+ h\tau \right) $$ in H1$$ {H}^1 $$ ‐norm, where h$$ h $$ and τ$$ \tau $$ denote the space and time step size. Then the unconditional optimal L2$$ {L}^2 $$ error and superclose result with order O(h2+τ2)$$ O\left({h}^2+{\tau}^2\right) $$ are deduced, and the unconditional optimal H1$$ {H}^1 $$ error is obtained with order O(h+τ2)$$ O\left(h+{\tau}^2\right) $$ by using interpolation theory. The final unconditional superconvergence result with order O(h2+τ2)$$ O\left({h}^2+{\tau}^2\right) $$ 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 0