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expanded mixed finite element method for second order hyperbolic equations

COMPUTERS & MATHEMATICS WITH APPLICATIONS 2019年第8期78卷 2560-2574页

作者： Wang, Keyan Wang, Qisheng Hengyang Normal Univ Sch Math & Stat Hengyang 421008 Hunan Peoples R China Wuyi Univ Sch Math & Computat Sci Jiangmen 529020 Guangdong Peoples R China

In this paper, an expanded mixed finite element method is developed to approximate the solution of second order hyperbolic equations. We discuss and theoretically prove the existence and uniqueness of solution for the semidiscrete formulation, and obtain the optimal order error estimates in the L-infinity(L-2) norm. In addition, we also construct the fully discrete approximations and provide the corresponding error analysis. Some numerical examples are presented to demonstrate the theoretical analysis. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Hyperbolic equations expanded mixed finite element method Error estimates

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Two-grid methods for semilinear time fractional reaction diffusion equations by expanded mixed finite element method

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APPLIED NUMERICAL MATHEMATICS 2020年 157卷 38-54页

作者： Li, Qingfeng Chen, Yanping Huang, Yunqing Wang, Yang South China Normal Univ Sch Math Sci Guangzhou 510631 Peoples R China Xiangtan Univ Sch Math & Computat Sci Hunan Key Lab Computat & Simulat Sci & Engn Xiangtan 411105 Peoples R China

In this paper, two-grid algorithms based on expanded mixed finite element method are presented for solving two-dimensional semilinear time fractional reaction-diffusion equations. To obtain the fully discrete scheme, the classical L1 scheme is considered in the time direction, and the expanded mixed finite element method is used to approximate spatial direction. Then the error estimates and stability of fully discrete scheme are derived. To linearize the nonlinear system, the two-grid method based on Newton iteration are constructed. The two-grid algorithms reduce the solution of the nonlinear fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithms save total computational cost. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Moreover, the numerical experiment is presented further confirm the theoretical results. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Two-grid method expanded mixed finite element method L1 scheme L-p error estimate Semilinear time fractional reaction diffusion equations

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Two-grid method of expanded mixed finite element Approximations for Parabolic Integro-differential Optimal Control Problems

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ACTA MATHEMATICAE APPLICATAE SINICA-ENGLISH SERIES 2024年 1-24页

作者： Chen, Yan-ping Zhou, Jian-wei Hou, Tian-liang Nanjing Univ Posts & Telecommun Sch Sci Nanjing 210023 Peoples R China Linyi Univ Sch Math & Stat Linyi 276005 Peoples R China Beihua Univ Sch Math & Stat Jilin 132013 Peoples R China

This paper aims to construct a two-grid scheme of fully discretized expanded mixed finite element methods for optimal control problems governed by parabolic integro-differential equations and discuss a priori error estimates. The state variables and co-state variables are discretized by the lowest order Raviart-Thomas mixed finite element, and the control variable is approximated by piecewise constant functions. The time derivative is discretized by the backward Euler method. Firstly, we define some new mixed elliptic projections and prove the corresponding error estimates which play an important role in subsequent convergence analysis. Secondly, we derive a priori error estimates for all variables. Thirdly, we present a two-grid scheme and analyze its convergence. In the two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid and the solution of a decoupled linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. At last, a numerical example is presented to verify the theoretical results.

关键词： linear parabolic integro-differential equations expanded mixed finite element method a priori error estimates two-grid superconvergence

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Convergence analysis of hybrid expanded mixed finite element method for elliptic equations

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2014年第10期68卷 1205-1219页

作者： Song, Huailing Jiang, Lijian Chen, Gaojie Hunan Univ Coll Math & Econometr Changsha 410082 Hunan Peoples R China

In this article, we utilize the approach of the variational characterization of the Lagrange multipliers in Cockburn and Gopalakrishnan (2005) and present the convergence analysis of hybrid expanded mixed finite element method for elliptic equations. The unknowns in the expanded mixed formulation can be expressed in terms of lifting operators. The analysis provides a novel perspective for error estimates of the hybrid expanded mixed finite element method. Numerical examples are presented to confirm the convergence analysis. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： expanded mixed finite element method Lagrange multipliers Lifting operators

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Probability-conservative simulation for Levy financial model by a mixed finite element method

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2022年第0期106卷 92-105页

作者： Chen, Jing Wang, Feng Chen, Huanzhen Shandong Normal Univ Sch Econ Jinan 250014 Peoples R China Shandong Univ Sci & Technol Coll Math & Syst Sci Qingdao 266590 Peoples R China Shandong Normal Univ Sch Math & Stat Jinan 250014 Peoples R China

We propose an expanded mixed finite element method (EMFEM) with sharp recognition for such intrinsic natures of a Markov process as the probability conservation, the Chapman-Kolmogorov equation, skewness and excess kurtosis for the fractional-order advection diffusion equation (FADE) oriented in financial engineering. We improve the result in [11] for purely fractional diffusion model by modifying the domain of one bilinear form so that it is well defined for the appearance of the advection term, and the result in [13] for second order elliptic models by using lower-order finite element spaces to reduce computation costs. The solvability and optimal convergence rates of the EMFEM are proved by showing the coerciveness and the inf-sup condition for the modified bilinear forms over the lower-order finite element spaces, and thus the systematically mathematical theory of the EMFEM for general FADEs are developed. Numerical experiments are conducted to confirm these theoretical findings.

关键词： Levy financial model Fractional advection diffusion equation Probability conservative simulation expanded mixed finite element method The inf-sup condition Solvability and convergence

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Convergence analysis of two-grid methods for second order hyperbolic equation

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2021年第1期2021卷 1-15页

作者： Wang, Keyan Wang, Qisheng Hengyang Normal Univ Sch Math & Stat Hengyang 421008 Hunan Peoples R China Wuyi Univ Sch Math & Computat Sci Jiangmen 529020 Guangdong Peoples R China

In this paper, a second-order hyperbolic equation is solved by a two-grid algorithm combined with the expanded mixed finite element method. The error estimate of the expanded mixed finite element method with discrete-time scheme is demonstrated. Moreover, we present a two-grid method and analyze its convergence. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h(1/2)). Finally, some numerical experiments are provided to illustrate the efficiency and accuracy of the proposed method.

关键词： Hyperbolic equation expanded mixed finite element method Two-grid algorithm Error estimate

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Two-grid methods for expanded mixed finite element approximations of semi-linear parabolic integro-differential equations

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APPLIED NUMERICAL MATHEMATICS 2018年 132卷 163-181页

作者： Hou, Tianliang Chen, Luoping Yang, Yin Beihua Univ Sch Math & Stat Jilin 132013 Jilin Peoples R China Southwest Jiaotong Univ Sch Math Chengdu 611756 Sichuan Peoples R China Xiangtan Univ Sch Math & Computat Sci Xiangtan 411105 Hunan Peoples R China

In this paper, we investigate a two grid discretization scheme for semilinear parabolic integro-differential equations by expanded mixed finite element methods. The lowest order Raviart-Thomas mixed finite element method and backward Euler method are used for spatial and temporal discretization respectively. Firstly, expanded mixed Ritz-Volterra projection is defined and the related a priori error estimates are proved. Secondly, a superconvergence property of the pressure variable for the fully discretized scheme is obtained. Thirdly, a two-grid scheme is presented to deal with the nonlinear part of the equation and a rigorous convergence analysis is given. It is shown that when the two mesh sizes satisfy h = H-2, the two grid method achieves the same convergence property as the expanded mixed finite element method. Finally, a numerical experiment is implemented to verify theoretical results of the two grid method. (C) 2018 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Semi-linear parabolic integro-differential equations expanded mixed finite element method A priori error estimates Two-grid scheme Superconvergence

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An expanded mixed finite element simulation for two-sided time-dependent fractional diffusion problem

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ADVANCES IN DIFFERENCE EQUATIONS 2018年第1期2018卷 1-15页

作者： Yuan, Qiong Chen, Huanzhen Shandong Normal Univ Sch Math & Stat Jinan Shandong Peoples R China

In this paper, we consider a time-dependent diffusion problem with two-sided Riemann-Liouville fractional derivatives. By introducing a fractional-order flux as auxiliary variable, we establish the saddle-point variational formulation, based on which we employ a locally conservative mixed finite element method to approximate the unknown function, its derivative and the fractional flux in space and use the backward Euler scheme to discrete the time derivative, and thus propose a fully discrete expanded mixed finite element procedure. We prove the well-posedness and the optimal order error estimates of the proposed procedure for a sufficiently smooth solution. Numerical experiments are presented to confirm our theoretical findings.

关键词： time-dependent fractional diffusion problem expanded mixed finite element method fully discrete scheme well-posedness error estimates

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Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2016年 296卷 480-498页

作者： Chen, Huanzhen Wang, Hong Shandong Normal Univ Coll Math Sci Jinan 250014 Peoples R China Univ S Carolina Dept Math Columbia SC 29208 USA

In this article we adopt the saddle-point theoretical framework to analyze the conservative fractional diffusion equations. By introducing a fractional-order flux as auxiliary variable, we establish the well-posedness of the saddle-point variational formulation. Based on the formulation we propose a locally-conservative expanded mixed finite element procedure to approximate the unknown, its derivative and the fractional flux directly, and prove the existence and uniqueness of the mixed finite element solution. By proving that the negative fractional derivative spaces are equivalent to the negative fractional-order Sobolev spaces we derive the approximation capability of the standard projection operator in fractional Sobolev spaces for non-smooth functions. This leads an optimal-order error estimates in terms of the right-hand side both for sufficiently smooth solution and non-smooth solution. Numerical experiments are included to confirm our theoretical findings. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Conservative fractional diffusion equation Saddle-point variational formulation Well-posedness expanded mixed finite element method Optimal-order error analysis Numerical experiments

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一类二阶线性抛物型方程的扩展混合有限元方法

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高等学校计算数学学报 2008年第3期30卷 234-249页

作者：刘洋李宏文宗川内蒙古大学理工学院数学系内蒙古工业大学管理学院

1引言近年来,很多计算数学工作者致力于偏微分方程混合有限元方法方面的研究,这个方法早期是于上个世纪六七十年代被几位工程师(Fraeijs de Veubeke,1965;Hellan,1967;Hermann,1967)用来解决solid continua问题而引入的,从那时起,混合... 详细信息

1引言近年来,很多计算数学工作者致力于偏微分方程混合有限元方法方面的研究,这个方法早期是于上个世纪六七十年代被几位工程师(Fraeijs de Veubeke,1965;Hellan,1967;Hermann,1967)用来解决solid continua问题而引入的,从那时起,混合有限元方法被广

关键词： parabolic equation expanded mixed finite element method uniqueness and existence stability error estimate

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