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Advances in Applied Mathematics and Mechanics 2018年第6期10卷 1440-1458页

作者： Xingfa Yang Yin Yang Yanping Chen Jie Liu State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body Hunan UniversityChangsha 410082HunanChina Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing&Information Processing of Ministry of EducationSchool of Mathematics and Computational ScienceXiangtan UniversityXiangtan 411105HunanChina School of Mathematical Sciences South China Normal UniversityGuangzhou 510631GuangdongChina Center for Research on Leading Technology of Special Equipment School of Mechanical and Electric EngineeringGuangzhou UniversityGuangzhou 510006GuangdongChina

In this paper,we study a class of nonlinear fractional integro-differential equations,the fractional derivative is described in the Caputo *** the properties of the Caputo derivative,we convert the fractional integro-differential equations into equivalent integral-differential equations of Volterra type with singular kernel,then we propose and analyze a spectral Jacobi-collocation approximation for nonlinear integro-differential equations of Volterra *** provide a rigorous error analysis for the spectral methods,which shows that both the errors of approximate solutions and the errors of approximate fractional derivatives of the solutions decay exponentially in L^(∞)-norm and weighted L^(2)-norm.

关键词： Spectral method nonlinear fractional derivative Volterra integro-differential equations Caputo derivative

Mesh Quality and More Detailed Error Estimates of Finite Element Method

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高等学校计算数学学报（英文版） 2017年第2期10卷 420-436页

作者： Yunqing Huang Liupeng Wang Nianyu Yi Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan Hunan 411105 China

In this paper,we study the role of mesh quality on the accuracy of linear finite element *** derive a more detailed error estimate,which shows explicitly how the shape and size of elements,and symmetry structure of mesh effect on the error of numerical *** computable parameters Ge and Gv are given to depict the cell geometry property and symmetry structure of the *** compare with the standard a priori error estimates,which only yield information on the asymptotic error behaviour in a global sense,our proposed error estimate considers the effect of local element geometry properties,and is thus more *** certain conditions,the traditional error estimates and supercovergence results can be derived from the proposed error ***,the estimators Ge and Gv are computable and thus can be used for predicting the variation of *** tests are presented to illustrate the performance of the proposed parameters Ge and Gv.

关键词： Finite element methods error estimation cell geometry mesh quality

A multigrid-reduction-in-time solver with a new two-level convergence for unsteady fractional Laplacian problems

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arXiv 2019年

作者： Yue, Xiaoqiang Pan, Kejia Zhou, Jie Weng, Zhifeng Shu, Shi Tang, Juan Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China School of Mathematics and Statistics HNP-LAMA Central South University Changsha410083 China School of Mathematical Sciences Fujian Province University Key Laboratory of Computation Science Huaqiao University Quanzhou362021 China

The multigrid-reduction-in-time (MGRIT) technique has proven to be successful in achieving higher run-time speedup by exploiting parallelism in time. The goal of this article is to develop and analyze a MGRIT algorithm, using FCF-relaxation with time-dependent time-grid propagators, to seek the finite element approximations of unsteady fractional Laplacian problems. The multigrid with line smoother proposed in [L. Chen, R. H. Nochetto, et al., Math. Comp. 85 (2016) 2583–2607] is chosen to be the spatial solver. Motivated by [B. S. Southworth, SIAM J. Matrix Anal. Appl. 40 (2019) 564–608], we provide a new temporal eigenvalue approximation property and then deduce a generalized two-level convergence theory which removes the previous unitary diagonalization assumption on the fine and coarse time-grid propagators required in [X. Q. Yue, S. Shu, et al., Comput. Math. Appl. 78 (2019) 3471–3484]. Numerical computations are included to confirm the theoretical predictions and demonstrate the sharpness of the derived convergence upper bound. Copyright © 2019, The Authors. All rights reserved.

关键词： Finite element method

SATURATION AND RELIABLE HIERARCHICAL A POSTERIORI MORLEY FINITE ELEMENT ERROR CONTROL

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Journal of computational Mathematics 2018年第6期36卷 833-844页

作者： Carsten Carstensen Dietmar Gallistl Yunqing Huang Humboldt-Universitat zu Berlin Institut fiir Mathematik Unter den Linden 6 10099 Berlin Germany Department of Applied Mathematics University of Twente P.O. Box 217 7500 AE Enschede Institute for Computational and Applied Mathematics and Hunan Key Laboratory for Computation & Simulation in Science & Engineering Xiangtan University Xiangtan 411105 China

This paper proves the saturation assumption for the nonconforming Morley finite ele- ment discretization of the biharmonic equation. This asserts that the error of the Morley approximation under uniform refinement is strictly reduced by a contraction factor smaller than one up to explicit higher-order data approximation terms. The refinement has at least to bisect any edge such as red refinement or 3-bisections on any triangle. This justifies a hierarchical error estimator for the Morley finite element method, which simply compares the discrete solutions of one mesh and its red-refinement. The related adaptive mesh-refining strategy performs optimally in numerical experiments. A remark for Crouzeix-Raviart nonconforming finite element error control is included.

关键词： Saturation Hierarchical error estimation Finite element Nonconforming Biharmonie Morley Kirchhoff plate Crouzeix-Raviart

Adaptive-multilevel BDDC algorithm for three-dimensional plane wave Helmholtz systems

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arXiv 2019年

作者： Peng, Jie Shu, Shi Wang, Junxian Zhong, Liuqiang School of Mathematical Sciences South China Normal University Guangzhou510631 China School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan411105 China

In this paper we are concerned with the weighted plane wave least-squares (PWLS) method for three-dimensional Helmholtz equations, and develop the multi-level adaptive BDDC algorithms for solving the resulting discrete system. In order to form the adaptive coarse components, the local generalized eigenvalue problems for each common face and each common edge are carefully designed. The condition number of the two-level adaptive BDDC preconditioned system is proved to be bounded above by a user-defined tolerance and a constant that is only dependent on the maximum number of faces and edges per subdomain and the number of subdomains sharing a common edge. The effectiveness of these algorithms is illustrated on benchmark problem. The numerical results show the robustness of our two-level adaptive BDDC algorithms with respect to the wave number, the number of subdomains and the mesh size, and appear that our multi-level adaptive BDDC algorithm can reduce the number of dofs at the coarse problem and can be used to solving large wave number problems efficiently. Copyright © 2019, The Authors. All rights reserved.

关键词： Elastic waves

A finite element method of the self-consistent field theory on general curved surfaces

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arXiv 2018年

作者： Wei, Huayi Xu, Ming Si, Wei Jiang, Kai School of Mathematics and Computational Science Xiangtan University Xiangtan Hunan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering

Block copolymers provide a wonderful platform in studying the soft condensed matter systems. Many fascinating ordered structures have been discovered in bulk and confined systems. Among various theories, the self-consistent field theory (SCFT) has been proven to be a powerful tool for studying the equilibrium ordered structures. Many numerical methods have been developed to solve the SCFT model. However, most of these focus on the bulk systems, and little work on the confined systems, especially on general curved surfaces. In this work, we developed a linear surface finite element method, which has a rigorous mathematical theory to guarantee numerical precsion, to study the self-assembled phases of block copolymers on general curved surfaces based on the SCFT. Furthermore, to capture the consistent surface for a given self-assembled pattern, an adaptive approach to optimize the size of the general curved surface has been proposed. To demonstrate the power of this approach, we investigate the self-assembled patterns of diblock copolymers on several distinct curved surfaces, including five closed surfaces and an unclosed surface. Numerical results illustrate the efficiency of the proposed method. The obtained ordered structures are consistent with the previous results on standard surfaces, such as sphere and torus. Certainly, the proposed numerical framework has the capability of studying the phase behaviors on general surfaces precisely. Copyright © 2018, The Authors. All rights reserved.

关键词： Mean field theory

High-order energy stable schemes of incommensurate phase-field crystal model

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arXiv 2018年

作者： Jiang, Kai Si, Wei School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan411105 China

This article focuses on the development of high-order energy stable schemes for the multi-length-scale incommensurate phase-field crystal model which is able to study the phase behavior of aperiodic structures. These high-order schemes based on the scalar auxiliary variable (SAV) and spectral deferred correction (SDC) approaches are suitable for the L2 gradient flow equation, i.e., the Allen-Cahn dynamic equation. Concretely, we propose a second-order Crank-Nicolson (CN) scheme of the SAV system, prove the energy dissipation law, and give the error estimate in the almost periodic function sense. Moreover, we use the SDC method to improve the computational accuracy of the SAV/CN scheme. Numerical results demonstrate the advantages of high-order numerical methods in numerical computations and show the influence of length-scales on the formation of ordered structures. Copyright © 2018, The Authors. All rights reserved.

关键词： Numerical methods

Fully Discrete Galerkin Finite Element Method for the Cubic Nonlinear Schr(o)dinger Equation

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高等学校计算数学学报（英文版） 2017年第3期10卷 671-688页

作者： Jianyun Wang Yunqing Huang School of Science Hunan University of Technology Zhuzhou 412007 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan 411105 China

This paper is concemed with numerical method for a two-dimensional timedependent cubic nonlinear Schr(o)dinger *** approximations are obtained by the Galerkin finite element method in space in conjunction with the backward Euler method and the Crank-Nicolson method in time,*** prove optimal L2 error estimates for two fully discrete schemes by using elliptic projection ***,a numerical example is provided to verify our theoretical results.

关键词： Finite element method nonlinear Schr(o)dinger equations backward Euler scheme Crank-Nicolson scheme

Recovery based finite element method for biharmonic equation in two dimensional

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arXiv 2018年

作者： Huang, Yunqing Wei, Huayi Yang, Wei Yi, Nianyu Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China

We design and numerically validate a recovery based linear finite element method for solving the biharmonic equation. The main idea is to replace the gradient operator ∇ on linear finite element space by G(∇) in the weak formulation of the biharmonic equation, where G is the recovery operator which recovers the piecewise constant function into the linear finite element space. By operator G, Laplace operator ∆ is replaced by ∇ · G(∇). Furthermore the boundary condition on normal derivative ∇u · n is treated by the boundary penalty method. The explicit matrix expression of the proposed method is also introduced. Numerical examples on uniform and adaptive meshes are presented to illustrate the correctness and effectiveness of the proposed *** Codes 65N30 Copyright © 2018, The Authors. All rights reserved.

关键词： Finite element method

An Improved Third-order CWENO-Z Scheme at critical points

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IOP Conference Series: Materials Science and engineering 2019年第3期569卷

作者： Shujiang Tang Mingjun Li Shuang Han School of mathematics and computational science Xiangtan University Xiangtan Hunan 411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan Hunan 411105 China

In this paper, by modifying the smoothness factor of the third-order CWENO scheme, we present the two-parameter CWENO-NZ3 scheme to improve accuracy at critical points. We selected some classical examples for numerical simulation, such as one-dimensional Sod shock tube, Shock-entropy wave interaction, Riemann problem with strong discontinuity and Rayleigh-Taylor instability problem. Numerically comparing with the classical third-order CWENO schemes, it is found that the improved schemes not only improve accuracy and resolution at the extreme points, but also reduce dissipation.

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