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optimal l~∞ error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions

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Science China Mathematics 2014年第10期57卷 2189-2214页

作者： WANG TingChun ZHAO XiaoFei School of Mathematics and Statistics Nanjing University of Information Science and Technology Department of Mathematics National University of Singapore

Due to the difficulty in obtaining the a priori estimate,it is very hard to establish the optimal point-wise error bound of a finite difference scheme for solving a nonlinear partial differential equation in high dimensions(2D or 3D).We here propose and analyze finite difference methods for solving the coupled GrossPitaevskii equations in two dimensions,which models the two-component Bose-Einstein condensates with an internal atomic Josephson *** methods which we considered include two conservative type schemes and two non-conservative type *** conservation laws and solvability of the schemes are *** the four proposed finite difference methods,we establish the optimal convergence rates for the error at the order of O(h^2+τ~2)in the l~∞-norm(i.e.,the point-wise error estimates)with the time stepτand the mesh size *** the standard techniques of the energy method,the key techniques in the analysis is to use the cut-off function technique,transformation between the time and space direction and the method of order *** the methods and results here are also valid and can be easily extended to the three-dimensional ***,numerical results are reported to confirm our theoretical error estimates for the numerical methods.

关键词： coupled Gross-Pitaevskii equations finite difference method solvability conservation laws pointwise convergence optimal error estimates

CONVERGENCE ANALYSIS OF NONCONFORMING QUADRILATERAL FINITE ELEMENT METHODS FOR NONLINEAR COUPLED SCHRODINGER-HELMHOLTZ EQUATIONS

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Journal of Computational Mathematics 2024年第4期42卷 979-998页

作者： Dongyang Shi Houchao Zhang School of Mathematics and Statistics Zhengzhou UniversityZhengzhou 450001China

The focus of this paper is on two novel linearized Crank-Nicolson schemes with nonconforming quadrilateral finite element methods(FEMs)for the nonlinear coupled Schrodinger-Helmholtz *** L^(2) and H^(1) estimates of orders O(h^(2)+τ^(2))and O(h^(2)+τ^(2))are derived respectively without any grid-ratio condition through the following two *** is that a time-discrete system is introduced to split the error into the temporal error and the spatial error,which leads to optimal temporal error estimates of order O(τ^(2))in L^(2) and the broken H^(1)-norms,as well as the uniform boundness of numerical solutions in L^(∞) *** other is that a novel projection is utilized,which can iron out the difficulty of the existence of the consistency *** leads to derive optimal spatial error estimates of orders O(h^(2))in L^(2)-norm and O(h)in the broken H^(1)-norm under the H^(2) regularity of the solutions for the time-discrete *** last,two numerical examples are provided to confirm the theoretical ***,h is the subdivision parameter,and τ is the time step.

关键词： Schrodinger-Helmholtz equations Nonconforming FEMs Linearized Crank-Nicolson scheme optimal error estimates

SUPERCONVERGENCE ANALYSIS OF A BDF-GALERKIN FEM FOR THE NONLINEAR KLEIN-GORDON-SCHRODINGER EQUATIONS WITH DAMPING MECHANISM

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Journal of Computational Mathematics 2023年第2期41卷 224-245页

作者： Dongyang Shi Houchao Zhang School of Mathematics and Statistics Zhengzhou UniversityZhengzhou 450001China

The focus of this paper is on a linearized backward differential formula(BDF)scheme with Galerkin FEM for the nonlinear Klein-Gordon-Schrödinger equations(KGSEs)with damping *** error estimates and superconvergence results are proved without any time-step restriction condition for the proposed *** proof consists of three ***,a temporal-spatial error splitting argument is employed to bound the numerical solution in certain strong ***,optimal error estimates are derived through a novel splitting technique to deal with the time derivative and some sharp estimates to cope with the nonlinear ***,by virtue of the relationship between the Ritz projection and the interpolation,as well as a so-called"lifting"technique,the superconvergence behavior of order O(h^(2)+τ^(2))in H^(1)-norm for the original variables are ***,a numerical experiment is conducted to confirm our theoretical ***,h is the spatial subdivision parameter,andτis the time step.

关键词： KGSEs with damping mechanism Linearized BDF Galerkin FEM optimal error estimates Superconvergence

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

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Numerical Mathematics(Theory,Methods and Applications) 2023年第2期16卷 323-347页

作者： Bhupen Deka Papri Roy Naresh Kumar Raman Kumar Department of Mathematics Indian Institute of Technology GuwahatiGuwahati 781039India

Weak Galerkin finite element method is introduced for solving wave equation with interface on weak Galerkin finite element space(Pk(K),P_(k−1)(∂K),[P_(k−1)(K)]^(2)).optimal order a priori error estimates for both space-discrete scheme and implicit fully discrete scheme are derived in L1(L2)*** method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal *** element algorithm presented here can contribute to a variety of hyperbolic problems where physical domain consists of heterogeneous media.

关键词： Wave equation heterogeneous medium finite element method weak Galerkin method semidiscrete and fully discrete schemes optimal error estimates

Linearized Transformed L1 Galerkin FEMs with Unconditional Convergence for Nonlinear Time Fractional Schr¨odinger Equations

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Numerical Mathematics(Theory,Methods and Applications) 2023年第2期16卷 348-369页

作者： Wanqiu Yuan Dongfang Li Chengjian Zhang School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China

A linearized transformed L1 Galerkin finite element method(FEM)is presented for numerically solving the multi-dimensional time fractional Schr¨odinger *** optimal error estimates of the fully-discrete scheme are *** error estimates are obtained by combining a new discrete fractional Gr¨onwall inequality,the corresponding Sobolev embedding theorems and some inverse *** the previous unconditional convergence results are usually obtained by using the temporal-spatial error spitting *** examples are presented to confirm the theoretical results.

关键词： optimal error estimates time fractional Schr¨odinger equations transformed L1 scheme discrete fractional Gr¨onwall inequality

Analysis of the local discontinuous Galerkin method with generalized fluxes for one-dimensional nonlinear convection-diffusion systems

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Science China Mathematics 2023年第11期66卷 2641-2664页

作者： Hongjuan Zhang Boying Wu Xiong Meng School of Mathematics Harbin Institute of TechnologyHarbin 150001China

In this paper,we present optimal error estimates of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear convection-diffusion *** upwind-biased flux with the adjustable numerical viscosity for the convective term is chosen based on the local characteristic decomposition,which is helpful in resolving discontinuities of degenerate parabolic equations without enforcing any limiting *** the diffusive term,a pair of generalized alternating fluxes is *** constructing and analyzing generalized Gauss-Radau projections with respect to different convective or diffusive terms,we derive optimal error estimates for nonlinear convection-diffusion systems with the symmetrizable flux Jacobian and fully nonlinear diffusive *** experiments including long time simulations,different boundary conditions and degenerate equations with discontinuous initial data are provided to demonstrate the sharpness of theoretical results.

关键词： local discontinuous Galerkin method nonlinear convection-diffusion systems generalized numerical fuxes optimal error estimates generalized Gauss-Radau projections

A Local Discontinuous Galerkin Method with Generalized Alternating Fluxes for 2D Nonlinear Schrödinger Equations

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Communications on Applied Mathematics and Computation 2022年第1期4卷 84-107页

作者： Hongjuan Zhang Boying Wu Xiong Meng School of Mathematics Harbin Institute of TechnologyHarbin 150001China School of Mathematics and Institute for Advanced Study in Mathematics Harbin Institute of TechnologyHarbin 150001China

In this paper,we consider the local discontinuous Galerkin method with generalized alter-nating numerical fluxes for two-dimensional nonlinear Schrödinger equations on Carte-sian *** generalized fluxes not only lead to a smaller magnitude of the errors,but can guarantee an energy conservative property that is useful for long time simulations in resolving *** virtue of generalized skew-symmetry property of the discontinuous Galerkin spatial operators,two energy equations are established and stability results con-taining energy conservation of the prime variable as well as auxiliary variables are *** derive optimal error estimates for nonlinear Schrödinger equations,an additional energy equation is constructed and two a priori error assumptions are ***,together with properties of some generalized Gauss-Radau projections and a suitable numerical initial condition,implies optimal order of k+*** experiments are given to demonstrate the theoretical results.

关键词： Local discontinuous Galerkin method Two-dimensional nonlinear Schrödinger equation Generalized alternating fluxes optimal error estimates

NONCONFORMING FINITE ELEMENT METHOD FOR NONLINEAR PARABOLIC EQUATIONS

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Journal of Systems Science & Complexity 2010年第2期23卷 395-402页

作者： Dongyang SHI Buying ZHANG Department of Mathematics Zhengzhou University Zhengzhou 450052 China. E&A College Hebei Normal University of Science & Technology Qinhuangdao 066004 China.

A nonconforming finite element method for the nonlinear parabolic equations is studied inthis *** convergence analysis is presented and the optimal error estimate in L^2(‖·‖_h)norm isobtained through Ritz projectio... 详细信息

关键词： Nonconforming finite element nonlinear parabolic equations optimal error estimates Ritz projection.

QUADRILATERAL FINITE ELEMENTS FOR PLANAR LINEAR ELASTICITY PROBLEM WITH LARGE LAME CONSTANT

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Journal of Computational Mathematics 1998年第4期16卷 357-366页

作者： Cheng, XL Huang, HC Zou, J Hangzhou Univ Dept Math Hangzhou Peoples R China Hong Kong Baptist Univ Dept Math Hong Kong Hong Kong Chinese Univ Hong Kong Dept Math Hong Kong Hong Kong

In this paper, we discuss the quadrilateral, finite element approximation to the two-dimensional linear elasticity problem associated with a homogeneous isotropic elastic material. The optimal convergence of the finite element method is proved for both the L-2-norm and energy-norm, and in particular, the convergence is uniform with respect to the Lame constant lambda. Also the performance of the scheme does not deteriorate as the material becomes nearly incompressible, Numerical experiments are given which are consistent with our theory.

关键词： planar linear elasticity optimal error estimates large Lame constant locking phenomenon