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A time two-grid algorithm for two-dimensional nonlinear time-fractional partial integro-differential equations

MATHEMATICS AND COMPUTERS IN SIMULATION 2024年 221卷 550-569页

作者： Mei, Yusha Cui, Mingrong Zeng, Fanhai Shandong Univ Sch Math Jinan 250100 Shandong Peoples R China

In this paper, a temporal second order two-grid difference scheme is proposed for the two-dimensional nonlinear time-fractional partial integro-differential equations with a weakly singular kernel. The first-order backward difference and L1 formula are used in the temporal direction to estimate the first level of time, the L2 - 1(sigma) formula and L1-type formula are used in the temporal direction for later time steps, and the central difference formula is used in the spatial directions. To improve the computational efficiency of nonlinear system, an efficient time two-grid algorithm is proposed. This algorithm firstly solves a nonlinear system on the coarse grid, and then the Lagrangian linear interpolation is applied on the coarse grid to estimate the function values on the fine grid. The stability and convergence of the two-grid difference scheme are analyzed by the energy method. The convergence order of the two-grid difference scheme is O(tau(2)(F) + tau(4)(C) + h(x)(2) + h(y)(2)), where tau(F) and tau(C) are the time step sizes of fine grid and coarse grid respectively, while h(x) and h(y) are the space step sizes. Numerical experiments show that the accuracy of the theoretical analysis and the efficiency of the two-grid algorithm.

关键词： Nonlinear fractional partial integro-differential equations Temporal second order scheme two-grid algorithm Finite difference method Stability and convergence

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A time two-grid algorithm for the two dimensional nonlinear fractional PIDE with a weakly singular kernel

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MATHEMATICS AND COMPUTERS IN SIMULATION 2022年 199卷 38-59页

作者： Wang, Furong Yang, Xuehua Zhang, Haixiang Wu, Lijiao Hunan Univ Technol Coll Sci Zhuzhou 412008 Peoples R China

The main aim of this paper is to solve the two-dimensional nonlinear fractional partial integro-differential equation (PIDE) with a weakly singular kernel by using the time two-grid finite difference (FD) algorithm. The second-order backward difference formula (BDF) and L1 scheme are used in time. The time two-grid algorithm is constructed to improve the solving efficiency of nonlinear systems. The Newton iteration is used to solve nonlinear discrete system on the coarse grid, and then we apply Lagrangian linear interpolation to attain the function value used in constructing the difference scheme on the fine grid. The second-order finite difference method (FDM) is used in space. The unconditional stability and convergence are attained for the two-grid fully discrete system. Numerical experiments show that the used CPU time for the presented two-grid numerical algorithm is lower than the general finite difference method for solving the nonlinear system. (c) 2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

关键词： Nonlinear partial integro-differential equation Time fractional derivative two-grid algorithm Finite difference method Stability and convergence

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Post-processing technique of two-grid algorithm for wave propagation with Debye polarization in nonlinear dielectric materials

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APPLIED NUMERICAL MATHEMATICS 2020年 157卷 405-418页

作者： Yao, Changhui Wei, Yifan Huang, Qiumei Zhengzhou Univ Sch Math & Stat Zhengzhou 450001 Peoples R China Chinese Acad Sci State Key Lab Space Weather Beijing 100190 Peoples R China Beijing Univ Technol Beijing Inst Sci & Engn Comp Beijing 100124 Peoples R China

In this paper, we develop the superconvergence analysis of the implicit second-order two-grid discrete scheme with the lowest Nedelec element for wave propagation with Debye Polarization in nonlinear Dielectric materials. Our main contribution will have two parts. On one hand, in order to overcome the difficulty of misconvergence of classical two-grid algorithm by the lowest Nedelec elements, we employ the Newton-type Taylor expansion at the superconvergent solutions for the nonlinear terms on coarse mesh, which is different from the classical numerical solution on the coarse mesh. On the other hand, we push the two-grid solution to high accuracy by the interpolation post-processing technique. Such a design can both improve the computational accuracy in spatial and decrease time consumption simultaneously. Based on this design, we can obtain the convergent rate O (tau(2) + h(2) + H-3), and the spatial convergence can be obtained by choosing the mesh size h = O (H-3/2). At last, one numerical experiment is illustrated to verify our theoretical results. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Wave propagation Nonlinear Nedelec element two-grid algorithm Post-processing

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A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile/immobile transport model

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NUMERICAL algorithmS 2020年第1期85卷 39-58页

作者： Qiu, Wenlin Xu, Da Guo, Jing Zhou, Jun Hunan Normal Univ Key Lab Comp & Stochast Math Sch Math & Stat Minist Educ Changsha 410081 Hunan Peoples R China

In this paper, we present a time two-grid algorithm based on the finite difference (FD) method for the two-dimensional nonlinear time-fractional mobile/immobile transport model. We establish the problem as a nonlinear fully discrete FD system, where the time derivative is discretized by the second-order backward difference formula (BDF) scheme, the Caputo fractional derivative is treated by means of L1 discretization formula, and the spatial derivative is approximated by the central difference formula. For solving the nonlinear FD system more efficiently, a time two-grid algorithm is proposed, which consists of two steps: first, the nonlinear FD system on a coarse grid is solved by nonlinear iterations;second, the Newton iteration is utilized to solve the linearized FD system on the fine grid. The stability and convergence inL(2)-norm are obtained for the two-grid FD scheme. Numerical results are consistent with the theoretical analysis. Meanwhile, numerical experiments show that the two-grid FD method is much more efficient than the general FD scheme for solving the nonlinear FD system.

关键词： two-grid algorithm two-dimensional Nonlinear time-fractional mobile immobile model Stability and convergence Numerical experiments

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The two-grid hybrid high-order method for the nonlinear strongly damped wave equation on polygonal mesh and its reduced-order model

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APPLIED NUMERICAL MATHEMATICS 2025年 210卷 1-24页

作者： Wang, Lu Tan, Youjun Feng, Minfu Sichuan Univ Coll Math Chengdu 610065 Peoples R China

This paper introduces the hybrid high-order (HHO) method for solving the nonlinear strongly damped wave equation. We comprehensively analyze the semi-discrete and fully-discrete implicit schemes, including energy and L2 norm, with convergence rates of m+ 1 and m+ 2 in space (m >= 0), respectively. In addition, we combine the two-grid algorithm (TGA) with the HHO method (TGA-HHO) to improve computational efficiency and analyze the TGA-HHO method. To improve the computational efficiency further, we combine the proper orthogonal decomposition (POD) technique with the TGA-HHO method (POD-TGA-HHO). Finally, we provide numerical examples to validate the effectiveness of the HHO, TGA-HHO, and POD-TGA-HHO algorithms.

关键词： Nonlinear wave equation Hybrid high-order method two-grid algorithm Proper orthogonal decomposition Semi-discrete scheme Fully-discrete scheme Polygonal mesh

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two-grid methods for nonlinear pseudo-parabolic integro-differential equations by finite element method

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2024年 168卷 174-189页

作者： Wang, Keyan Hengyang Normal Univ Sch Math & Stat Hengyang 421008 Hunan Peoples R China

In this paper, two efficient two-grid finite element algorithms are proposed for solving two-dimensional nonlinear pseudo-parabolic integro-differential equations. Firstly, we obtain optimal error estimates of the fully discrete finite element method using a temporal-spatial error splitting technique in H-1 and L-p norms. Then the two-grid technique to improve computation efficiency of the proposed finite element method. Error estimates in H-1 and nonlinear L-p norms of two-grid solutions are presented. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Finally, numerical examples are provided to support our theoretical results and demonstrate the effectiveness of these methods.

关键词： Pseudo-parabolic integro-differential equations Finite element method two-grid algorithm Error estimate

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two-grid finite element methods for nonlinear time-fractional parabolic equations

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NUMERICAL algorithmS 2022年第2期90卷 709-730页

作者： Zhou, Jie Yao, Xing Wang, Wansheng Xiangtan Univ Sch Math & Computat Sci Hunan Key Lab Computat & Simulat Sci & Engn Xiangtan 411105 Hunan Peoples R China Xiangtan Univ Sch Math & Computat Sci Xiangtan 411105 Hunan Peoples R China Shanghai Normal Univ Dept Math Shanghai 200234 Peoples R China

In this paper, an efficient two-grid finite element algorithm is proposed for solving time-fractional nonlinear parabolic equations. We first obtain the stability and error estimates of the standard fully discrete L1 finite element method by using the Gronwall inequalities in L-2 and H-1 norms. Based on the standard method, we design the corresponding two-grid algorithm and analyze its stability and error estimates. It is shown that this algorithm is as stable as the standard fully discrete finite element algorithm, and can achieve the same accuracy as the standard algorithm if the coarse grid size H and the fine grid size h satisfy H = O(hr-1/r)(r >= 1+ d/2, where d = 1,2,3). The theoretical results are illustrated by applying the proposed method to an numerical example.

关键词： Nonlinear time-fractional parabolic equation two-grid algorithm Finite element method L1 method Gronwall inequality Stability Error estimate

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Analysis of finite element two-grid algorithms for two-dimensional nonlinear Schrodinger equation with wave operator

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2021年 397卷 113647-113647页

作者： Hu, Hanzhang Chen, Yanping Jiaying Univ Sch Math Meizhou 514015 Guangdong Peoples R China South China Normal Univ Sch Math Sci Guangzhou 510631 Guangdong Peoples R China

two-grid algorithms based on two conservative and implicit finite element methods are studied for two-dimensional nonlinear Schrodinger equation with wave operator. The existence and uniqueness of their solutions and the conservative laws are established for both schemes. To linearize the fully discrete nonlinear coupling problem, the original problem is decomposed into two equivalent nonlinear coupled hyperbolic-parabolic equations. algorithm 1 has three steps based on one Newton iteration on the fine grid and further correction on coarse grid, while algorithm 2 has three steps based on two Newton iterations on the fine grid. Optimal order L-p error estimations of the two-grid algorithms are conducted in detail by optimal order L-p error estimates of finite element methods without any time-step size conditions. Both theoretically and numerically are shown that the coarse space can be extremely coarse, with no loss in the order of accuracy, and the two-grid algorithms still achieve the optimal convergence order when the mesh sizes satisfy H = O(h(1/3)) for algorithm 1 and H = O(h(1/4) ) for algorithm 2. (C) 2021 Elsevier B.V. All rights reserved.

关键词： Schrodinger equation with wave operator two-grid algorithm Conservative Convergence Finite element method

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A two-grid method with backtracking for the mixed Stokes/Darcy model

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JOURNAL OF NUMERICAL MATHEMATICS 2021年第1期29卷 39-46页

作者： Du, Guangzhi Zuo, Liyun Shandong Normal Univ Sch Math & Stat Jinan 250014 Peoples R China Univ Jinan Sch Math Sci Jinan 250022 Peoples R China

In this paper, a two-grid method with backtracking is proposed and investigated for the mixed Stokes/Darcy system which describes a fluid low coupled with a porous media low. Based on the classical two-grid method [15], a coarse mesh correction is carried out to derive optimal error bounds for the velocity field and the piezometric head in L-2 norm. Finally, results of numerical experiments are provided to support the theoretical results.

关键词： Stokes equations Darcy's law inite element method two-grid algorithm backtracking technique

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A mixed element computing scheme for a semilinear fourth-order problem

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COMPUTATIONAL & APPLIED MATHEMATICS 2021年第4期40卷 1-15页

作者： Song, Shicang Xie, Shuqi Zhengzhou Univ Sch Math & Stat Zhengzhou 450001 Peoples R China

A mixed finite element method is developed for a semilinear fourth-order elliptic boundary value problem. The existence and uniqueness of the solutions to the mixing problem are proved by constructing an auxiliary problem. Moreover, a computational format is provided for numerical calculation, the convergence of this problem is proved under some hypothetical conditions. The optimal error estimate of the semilinear fourth-order elliptic boundary value problem is analyzed by a two-grid method. Finally, some numerical examples are provided to support the theoretical analysis.

关键词： Semilinear fourth-order equation Mixed element two-grid algorithm Error estimate

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