Close Search Advanced
Journals
Resources
About Us
Open Access

Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods

Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods
Preview
Add to basket

Year:    2009

Author:    Yanping Chen, Peng Luan, Zuliang Lu

Advances in Applied Mathematics and Mechanics, Vol. 1 (2009), Iss. 6 : pp. 830–844

Abstract

In this paper, we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method. We use two Newton iterations on the fine grid in our methods. Firstly, we solve an original nonlinear problem on the coarse nonlinear grid, then we use Newton iterations on the fine grid twice. The two-grid idea is from Xu$'$s work [SIAM J. Numer. Anal., 33 (1996), pp. 1759-1777] on standard finite method. We also obtain the error estimates for the algorithms of the two-grid method. It is shown that the algorithm achieves asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy $h=\mathcal{O}(H^{(4k+1)/(k+1)})$.

Submit Article

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.09-m09S09

Advances in Applied Mathematics and Mechanics, Vol. 1 (2009), Iss. 6 : pp. 830–844

Published online:    2009-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    15

Keywords:    Nonlinear parabolic equations two-grid scheme expanded mixed finite element methods Gronwall's Lemma.

Author Details

Yanping Chen

Peng Luan

Zuliang Lu

  1. Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations

    Chen, Chuanjun | Li, Kang | Chen, Yanping | Huang, Yunqing

    Advances in Computational Mathematics, Vol. 45 (2019), Iss. 2 P.611

    https://doi.org/10.1007/s10444-018-9628-2 [Citations: 67]

  2. A new expanded mixed method for parabolic integro-differential equations

    Liu, Yang | Fang, Zhichao | Li, Hong | He, Siriguleng | Gao, Wei

    Applied Mathematics and Computation, Vol. 259 (2015), Iss. P.600

    https://doi.org/10.1016/j.amc.2015.02.081 [Citations: 6]

  3. A two-grid MMOC finite element method for nonlinear variable-order time-fractional mobile/immobile advection–diffusion equations

    Chen, Chuanjun | Liu, Huan | Zheng, Xiangcheng | Wang, Hong

    Computers & Mathematics with Applications, Vol. 79 (2020), Iss. 9 P.2771

    https://doi.org/10.1016/j.camwa.2019.12.008 [Citations: 68]

  4. Two-grid method for the two-dimensional time-dependent Schrödinger equation by the finite element method

    Tian, Zhikun | Chen, Yanping | Huang, Yunqing | Wang, Jianyun

    Computers & Mathematics with Applications, Vol. 77 (2019), Iss. 12 P.3043

    https://doi.org/10.1016/j.camwa.2019.01.030 [Citations: 14]

  5. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods

    Wang, Keyan

    Open Mathematics, Vol. 22 (2024), Iss. 1

    https://doi.org/10.1515/math-2024-0048 [Citations: 0]

  6. Two-grid fully discrete finite element algorithms on temporal graded meshes for nonlinear multi-term time-fractional diffusion equations with variable coefficient

    Li, Kang | Tan, Zhijun

    Communications in Nonlinear Science and Numerical Simulation, Vol. 125 (2023), Iss. P.107360

    https://doi.org/10.1016/j.cnsns.2023.107360 [Citations: 3]

  7. A Novel Characteristic Expanded Mixed Method for Reaction-Convection-Diffusion Problems

    Liu, Yang | Li, Hong | Gao, Wei | He, Siriguleng | Fang, Zhichao

    Journal of Applied Mathematics, Vol. 2013 (2013), Iss. P.1

    https://doi.org/10.1155/2013/683205 [Citations: 2]

  8. Two-grid methods of expanded mixed finite-element solutions for nonlinear parabolic problems

    Chen, Yanping | Wang, Yang | Huang, Yunqing | Fu, Longxia

    Applied Numerical Mathematics, Vol. 144 (2019), Iss. P.204

    https://doi.org/10.1016/j.apnum.2019.04.015 [Citations: 17]

  9. A Two-Grid Finite Element Method for a Second-Order Nonlinear Hyperbolic Equation

    Chen, Chuanjun | Liu, Wei | Zhao, Xin

    Abstract and Applied Analysis, Vol. 2014 (2014), Iss. P.1

    https://doi.org/10.1155/2014/803615 [Citations: 0]

  10. A New Positive Definite Expanded Mixed Finite Element Method for Parabolic Integrodifferential Equations

    Liu, Yang | Li, Hong | Wang, Jinfeng | Gao, Wei | Yuan, Jinyun

    Journal of Applied Mathematics, Vol. 2012 (2012), Iss. 1

    https://doi.org/10.1155/2012/391372 [Citations: 4]

  11. A fully discrete two-grid finite element method for nonlinear hyperbolic integro-differential equation

    Tan, Zhijun | Li, Kang | Chen, Yanping

    Applied Mathematics and Computation, Vol. 413 (2022), Iss. P.126596

    https://doi.org/10.1016/j.amc.2021.126596 [Citations: 4]

  12. A Coupling Method of New EMFE and FE for Fourth-Order Partial Differential Equation of Parabolic Type

    Liu, Yang | Li, Hong | Fang, Zhichao | He, Siriguleng | Wang, Jinfeng

    Advances in Mathematical Physics, Vol. 2013 (2013), Iss. P.1

    https://doi.org/10.1155/2013/787891 [Citations: 3]

  13. A New Expanded Mixed Finite Element Method for Parabolic Integro-Differential Equations with Nonlinear Memory

    Kamil Naeemah, Ali | Oda Al-Humedi, Hameeda

    IOP Conference Series: Materials Science and Engineering, Vol. 928 (2020), Iss. 4 P.042020

    https://doi.org/10.1088/1757-899X/928/4/042020 [Citations: 0]

  14. Superconvergence of Mixed Methods for Optimal Control Problems Governed by Parabolic Equations

    Xing, Xiaoqing | Chen, Yanping

    Advances in Applied Mathematics and Mechanics, Vol. 3 (2011), Iss. 4 P.401

    https://doi.org/10.4208/aamm.10-m1006 [Citations: 7]

  15. $$L^p$$ L p Error Estimates of Two-Grid Method for Miscible Displacement Problem

    Chen, Yanping | Zeng, Jiaoyan | Zhou, Jie

    Journal of Scientific Computing, Vol. 69 (2016), Iss. 1 P.28

    https://doi.org/10.1007/s10915-016-0187-8 [Citations: 19]

  16. TGMFE algorithm combined with some time second-order schemes for nonlinear fourth-order reaction diffusion system

    Yin, Baoli | Liu, Yang | Li, Hong | He, Siriguleng | Wang, Jinfeng

    Results in Applied Mathematics, Vol. 4 (2019), Iss. P.100080

    https://doi.org/10.1016/j.rinam.2019.100080 [Citations: 4]

  17. Analysis of two-grid method for semi-linear elliptic equations by new mixed finite element scheme

    Weng, Zhifeng | Feng, Xinlong | Zhai, Shuying

    Applied Mathematics and Computation, Vol. 219 (2013), Iss. 9 P.4826

    https://doi.org/10.1016/j.amc.2012.10.108 [Citations: 4]

  18. A New Two-Grid Method for Expanded Mixed Finite Element Solution of Nonlinear Reaction Diffusion Equations

    Liu, Shang | Chen, Yanping

    Advances in Applied Mathematics and Mechanics, Vol. 9 (2017), Iss. 3 P.757

    https://doi.org/10.4208/aamm.2015.m1370 [Citations: 7]

  19. A two-grid stabilized mixed finite element method for semilinear elliptic equations

    Weng, Zhifeng | Feng, Xinlong | Liu, Demin

    Applied Mathematical Modelling, Vol. 37 (2013), Iss. 10-11 P.7037

    https://doi.org/10.1016/j.apm.2013.02.016 [Citations: 10]

  20. A posteriori error estimates of two-grid finite volume element methods for nonlinear elliptic problems

    Chen, Chuanjun | Chen, Yanping | Zhao, Xin

    Computers & Mathematics with Applications, Vol. 75 (2018), Iss. 5 P.1756

    https://doi.org/10.1016/j.camwa.2017.11.035 [Citations: 8]

  21. A Two-Grid Algorithm of the Finite Element Method for the Two-Dimensional Time-Dependent Schrödinger Equation

    Wang, Jianyun | Zhong, Zixin | Tian, Zhikun | Liu, Ying

    Mathematics, Vol. 12 (2024), Iss. 5 P.726

    https://doi.org/10.3390/math12050726 [Citations: 0]

  22. A New Expanded Mixed Element Method for Convection-Dominated Sobolev Equation

    Wang, Jinfeng | Liu, Yang | Li, Hong | Fang, Zhichao

    The Scientific World Journal, Vol. 2014 (2014), Iss. P.1

    https://doi.org/10.1155/2014/297825 [Citations: 0]

  23. A Two‐Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations

    Chen, Chuanjun | Liu, Wei | Zhang, Xinguang

    Abstract and Applied Analysis, Vol. 2012 (2012), Iss. 1

    https://doi.org/10.1155/2012/391918 [Citations: 5]

  24. A two-grid virtual element method for nonlinear variable-order time-fractional diffusion equation on polygonal meshes

    Gu, Qiling | Chen, Yanping | Zhou, Jianwei | Huang, Yunqing

    International Journal of Computer Mathematics, Vol. 100 (2023), Iss. 11 P.2124

    https://doi.org/10.1080/00207160.2023.2263589 [Citations: 1]

  25. A NEW CHARACTERISTIC EXPANDED MIXED METHOD FOR SOBOLEV EQUATION WITH CONVECTION TERM

    LIU, YANG | LI, HONG | HE, SIRIGULENG | FANG, ZHICHAO | WANG, JINFENG

    International Journal of Modeling, Simulation, and Scientific Computing, Vol. 05 (2014), Iss. 01 P.1350015

    https://doi.org/10.1142/S1793962313500153 [Citations: 1]

  26. A two-grid mixed finite element method for a nonlinear fourth-order reaction–diffusion problem with time-fractional derivative

    Liu, Yang | Du, Yanwei | Li, Hong | Li, Jichun | He, Siriguleng

    Computers & Mathematics with Applications, Vol. 70 (2015), Iss. 10 P.2474

    https://doi.org/10.1016/j.camwa.2015.09.012 [Citations: 126]