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Unconditional optimal error estimates of linearized backward Euler Galerkin FEMs for nonlinear Schrödinger-Helmholtz equations

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Abstract

In this paper, we establish unconditionally optimal error estimates for linearized backward Euler Galerkin finite element methods (FEMs) applied to nonlinear Schrödinger-Helmholtz equations. By using the temporal-spatial error splitting techniques, we split the error between the exact solution and the numerical solution into two parts which are called the temporal error and the spatial error. First, by introducing a time-discrete system, we prove the uniform boundedness for the solution of this time-discrete system in some strong norms and derive error estimates in temporal direction. Second, by the above achievements, we obtain the boundedness of the numerical solution in \(L^{\infty }\)-norm. Then, the optimal L2 error estimates for r-order FEMs are derived without any restriction on the time step size. Numerical results in both two- and three-dimensional spaces are provided to illustrate the theoretical predictions and demonstrate the efficiency of the methods.

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Funding

This work was supported by the Natural Science Foundation of China (NSFC) under grants 11871393 and 61663043, the key project of the International Science and Technology Cooperation Program of Shaanxi Research & Development Plan (2019KWZ-08), and the Doctoral Foundation of Yunnan Normal University (No. 00800205020503093) and the Scientific Research Program Funded by Yunnan Provincial Education Department under grant no. 2019J0076.

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  1. Department of Mathematics, Yunnan Normal University, Kunming, 650500, People’s Republic of China

    Yun-Bo Yang

  2. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Yao-Lin Jiang

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  1. Yun-Bo Yang
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Correspondence to Yun-Bo Yang.

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Yang, YB., Jiang, YL. Unconditional optimal error estimates of linearized backward Euler Galerkin FEMs for nonlinear Schrödinger-Helmholtz equations. Numer Algor 86, 1495–1522 (2021). https://doi.org/10.1007/s11075-020-00942-5

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