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A multilevel correction method for Interior Transmission Eigenvalue Problem

JOURNAL OF SCIENTIFIC COMPUTING 2017年第2期72卷 586-604页

作者： Xie, Hehu Wu, Xinming Chinese Acad Sci Acad Math & Syst Sci LSEC ICMSEC Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China Fudan Univ Shanghai Key Lab Contemporary Appl Math Sch Math Sci Shanghai 200433 Peoples R China

In this paper, we give a numerical analysis for the transmission eigenvalue problem by the finite element method. A type of multilevel correction method is proposed to solve the transmission eigenvalue problem. The multilevel correction method can transform the transmission eigenvalue solving in the finest finite element space to a sequence of linear problems and some transmission eigenvalue solving in a very low dimensional spaces. Since the main computational work is to solve the sequence of linear problems, the multilevel correction method improves the overfull efficiency of the transmission eigenvalue solving. Some numerical examples are provided to validate the theoretical results and the efficiency of the proposed numerical scheme.

关键词： Transmission eigenvalue problem Finite element method Error estimates multilevel correction method

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Adaptive multilevel correction method for Finite Element Approximations of Elliptic Optimal Control Problems

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JOURNAL OF SCIENTIFIC COMPUTING 2017年第2期72卷 820-841页

作者： Gong, Wei Xie, Hehu Yan, Ningning Chinese Acad Sci Acad Math & Syst Sci Inst Computat Math NCMISLSEC Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China Chinese Acad Sci Acad Math & Syst Sci Inst Syst Sci NCMISLSEC Beijing 100190 Peoples R China

In this paper we propose an adaptive multilevel correction scheme to solve optimal control problems discretized with finite element method. Different from the classical adaptive finite element method (AFEM for short) applied to optimal control which requires the solution of the optimization problem on new finite element space after each mesh refinement, with our approach we only need to solve two linear boundary value problems on current refined mesh and an optimization problem on a very low dimensional space. The linear boundary value problems can be solved with well-established multigrid method designed for elliptic equation and the optimization problems are of small scale corresponding to the space built with the coarsest space plus two enriched bases. Our approach can achieve the similar accuracy with standard AFEM but greatly reduces the computational cost. Numerical experiments demonstrate the efficiency of our proposed algorithm.

关键词： Optimal control problems Elliptic equation Control constraints A posteriori error estimates Adaptive finite element method multilevel correction method

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A multilevel correction method FOR OPTIMAL CONTROLS OF ELLIPTIC EQUATIONS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2015年第5期37卷 A2198-A2221页

作者： Gong, Wei Xie, Hehu Yan, Ningning Chinese Acad Sci Acad Math & Syst Sci Inst Computat Math LSEC Beijing 100190 Peoples R China Chinese Acad Sci Acad Math & Syst Sci Inst Computat Math NCMISLSEC Beijing 100190 Peoples R China

We propose in this paper a multilevel correction method to solve optimal control problems constrained by elliptic equations with the finite element method. In this scheme, solving an optimization problem on the finest finite element space is transformed into a series of solutions of linear boundary value problems by the multigrid method on multilevel meshes and a series of solutions of optimization problems on the coarsest finite element space. Our proposed scheme, instead of solving a large scale optimization problem in the finest finite element space, solves only a series of linear boundary value problems and the optimization problems in a very low dimensional finite element space, and thus can improve the overall efficiency of the solution of optimal control problems governed by PDEs.

关键词： optimal control problems elliptic equation control constraints finite element method multilevel correction method

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multilevel Local Defect-correction method for Nonsymmetric Eigenvalue Problems

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JOURNAL OF SCIENTIFIC COMPUTING 2022年第3期92卷 1-26页

作者： Xu, Fei Huang, Qiumei Dai, Haishen Ma, Hongkun Beijing Univ Technol Fac Sci Sch Math Beijing 100124 Peoples R China Zhuhai Huafa Investment Holdings Grp Co Ltd Hengqin 519000 Peoples R China

In this study, a multilevel local defect-correction method is designed for solving nonsymmetric eigenvalue problems. The main feature of our approach is the transformation of the nonsymmetric eigenvalue problems into several symmetric boundary value problems defined in a multilevel finite element space sequence and some low-dimensional nonsymmetric eigenvalue problems defined in a specially designed correction space. Moreover, the symmetric boundary value problems involved in our algorithm are solved by the local defect-correction strategy that divides the computing domain into small-scale subdomains. Since solving the high-dimensional nonsymmetric eigenvalue problems is avoided which is quite time-consuming compared with that of solving boundary value problems, the presented algorithm greatly improves the solving efficiency for nonsymmetric eigenvalue problems. Rigorous theoretical analysis and several numerical experiments are given to demonstrate the efficiency of our algorithm.

关键词： Nonsymmetric eigenvalue problems Local defect-correction method multilevel correction method

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multilevel correction goal-oriented adaptive finite element method for semilinear elliptic equations

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APPLIED NUMERICAL MATHEMATICS 2022年 172卷 224-241页

作者： Xu, Fei Huang, Qiumei Yang, Huiting Ma, Hongkun Beijing Univ Technol Fac Sci Beijing Inst Sci & Engn Comp Beijing 100124 Peoples R China Zhuhai Huafa Investment Holdings Grp Co Ltd Hengqin 519000 Peoples R China Sun Yat Sen Univ Business Sch Guangzhou 510275 Peoples R China

In this study, a multilevel correction-type goal-oriented adaptive finite element method is designed for semilinear elliptic equations. Concurrently, the corresponding convergence property is theoretically proved. In the novel goal-oriented adaptive finite element method, only a linearized primal equation and a linearized dual equation are required to be solved in each adaptive finite element space. To ensure convergence, the approximate solution of the primal equation was corrected by solving a small-scale semilinear elliptic equation after the central solving process in each adaptive finite element space. Since solving of the large-scale semilinear elliptic equations is avoided and the goal-oriented technique is absorbed, there has been a significant improvement in the solving efficiency for the goal functional of semilinear elliptic equations. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Adaptive finite element method Goal-oriented multilevel correction method Convergence

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multilevel correction adaptive finite element method for solving nonsymmetric eigenvalue problems

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2021年第2期47卷 1-30页

作者： Xu, Fei Yue, Meiling Zheng, Bin Beijing Univ Technol Beijing Inst Sci & Engn Comp Fac Sci Beijing 100124 Peoples R China Beijing Technol & Business Univ Sch Math & Stat Beijing 100048 Peoples R China

Large-scale nonsymmetric eigenvalue problems are common in various fields of science and engineering computing. However, their efficient handling is challenging, and research on their solution algorithms is limited. In this study, a new multilevel correction adaptive finite element method is designed for solving nonsymmetric eigenvalue problems based on the adaptive refinement technique and multilevel correction scheme. Different from the classical adaptive finite element method, which requires solving a nonsymmetric eigenvalue problem in each adaptive refinement space, our approach requires solving a symmetric linear boundary value problem in the current refined space and a small-scale nonsymmetric eigenvalue problem in an enriched correction space. Since it is time-consuming to solve a large-scale nonsymmetric eigenvalue problem directly in adaptive spaces, the proposed method can achieve nearly the same efficiency as the classical adaptive algorithm when solving the symmetric linear boundary value problem. In addition, the corresponding convergence and optimal complexity are verified theoretically and demonstrated numerically.

关键词： Nonsymmetric eigenvalue problems Adaptive finite element method multilevel correction method Convergence and optimality complexity

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On accelerating a multilevel correction adaptive finite element method for Kohn-Sham equation

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JOURNAL OF COMPUTATIONAL PHYSICS 2023年 472卷

作者： Hu, Guanghui Xie, Hehu Xu, Fei Univ Macau Fac Sci & Technol Dept Math & Guangdong Hong Kong Macao Joint Lab Da Macau Peoples R China Zhuhai UM Sci & Technol Res Inst Zhuhai Guangdong Peoples R China Chinese Acad Sci Acad Math & Syst Sci LSEC ICMSEC Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China Beijing Univ Technol Fac Sci Beijing 100124 Peoples R China

Based on the numerical method proposed in Hu et al. (2018) [22] for Kohn-Sham equation, further improvement on the efficiency is obtained in this paper by i). designing a numerical method with the strategy of separately handling the nonlinear Hartree potential and exchange-correlation potential, and ii). parallelizing the algorithm in an eigenpairwise approach. The feasibility of two approaches is analyzed in detail, and the new algorithm is described completely. Compared with previous results, a significant improvement of numerical efficiency can be observed from plenty of numerical experiments, which make the new method more suitable for the practical problems.(c) 2022 Elsevier Inc. All rights reserved.

关键词： Kohn-Sham equation multilevel correction method Finite element method Separately handing nonlinear terms Parallel computing

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A cascadic multigrid method for nonsymmetric eigenvalue problem

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APPLIED NUMERICAL MATHEMATICS 2019年 146卷 55-72页

作者： Yue, Meiling Xie, Hehu Xie, Manting Beijing Technol & Business Univ Sch Math & Stat Beijing 100048 Peoples R China Chinese Acad Sci Acad Math & Syst Sci Inst Computat Math LSECNCMIS Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China Tianjin Univ Ctr Appl Math Tianjin 300072 Peoples R China

In this paper, a cascadic multigrid method is proposed to solve nonsymmetric eigenvalue problems. Based on the multilevel correction method, the proposed method transforms a nonsymmetric eigenvalue problem solving on the finest finite element space to linear smoothing steps on a sequence of multilevel finite element spaces and some nonsymmetric eigenvalue problems solving on a very low dimensional space. Choosing the sequence of finite element spaces and the number of smoothing steps appropriately, we obtain the optimal convergence rate with the optimal computing complexity. Some numerical examples are provided to validate the theoretical results and the efficiency of this proposed scheme. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Nonsymmetric eigenvalue problem Cascadic multigrid multilevel correction method Finite element method

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Local and parallel multigrid method for semilinear elliptic equations

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APPLIED NUMERICAL MATHEMATICS 2021年 162卷 20-34页

作者： Xu, Fei Huang, Qiumei Jiang, Kun Ma, Hongkun Beijing Univ Technol Fac Sci Beijing Inst Sci & Engn Comp Beijing 100124 Peoples R China Zhuhai Huafa Investment Holdings Grp Co Ltd Hengqin 519000 Peoples R China Sun Yat Sen Univ Business Sch Guangzhou 510275 Guangdong Peoples R China

This paper presents a new type of local and parallel multigrid method to solve semilinear elliptic equations. The proposed method does not directly solve the semilinear elliptic equations on each layer of the multigrid mesh sequence, but transforms the semilinear elliptic equations into several linear elliptic equations on the multigrid mesh sequence and some low-dimensional semilinear elliptic equations on the coarsest mesh. Furthermore, the local and parallel strategy is used to solve the involved linear elliptic equations. Since solving large-scale semilinear elliptic equations in fine space, which can be fairly time-consuming, is avoided, the proposed local and parallel multigrid scheme will significantly improve the solving efficiency for the semilinear elliptic equations. Besides, compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only requires the Lipschitz continuation property of the nonlinear term. We make a rigorous theoretical analysis of the presented local and parallel multigrid scheme, and propose some numerical experiments to support the theory. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Semilinear elliptic equations Local and parallel multilevel correction method Multigrid method

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A novel adaptive finite element method for the ground state solution of Bose-Einstein condensates

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APPLIED MATHEMATICS AND COMPUTATION 2020年 385卷

作者： Xu, Fei Huang, Qiumei Wang, Min Ma, Hongkun Beijing Univ Technol Fac Sci Beijing Inst Sci & Engn Comp Beijing 100124 Peoples R China Sun Yat Sen Univ Sch Business Guangzhou 510275 Guangdong Peoples R China Zhuhai Huafa Investment Holdings Co Ltd Zhuhai 519031 Guangdong Peoples R China

In this study we propose a novel adaptive finite element method for the ground state solution of Bose-Einstein Condensates (BEC). Different from the classical adaptive scheme applied to BEC which needs to solve a nonlinear eigenvalue model directly on each adaptive finite element space, our scheme requires to solve a linear elliptic boundary value problem on current adaptive space and a nonlinear eigenvalue model on a quite low-dimensional correction space. Further, the linear elliptic boundary value problem is solved by adaptive multigrid iterations. Since there is no nonlinear eigenvalue model to be solved directly on the adaptive spaces, the solving efficiency can be improved evidently. In addition, the convergence analysis of the proposed adaptive algorithm are derived numerically and theoretically. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Bose-Einstein condensates Adaptive multigrid method multilevel correction method Convergence Optimal complexity

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