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- Title
Conservative EQ1rot nonconforming FEM for nonlinear Schrödinger equation with wave operator.
- Authors
Wang, Lingli; Li, Meng; Peng, Shanshan
- 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<ϵ≪1$$ 0<\epsilon \ll 1 $$. 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.
- Subjects
NONLINEAR wave equations; NONLINEAR Schrodinger equation; OPERATOR equations; NUMERICAL analysis; CONSERVATION of mass; SCHRODINGER equation
- Publication
Numerical Methods for Partial Differential Equations, 2024, Vol 40, Issue 1, p1
- ISSN
0749-159X
- Publication type
Article
- DOI
10.1002/num.23057