A TWO-GRID FINITE ELEMENT APPROXIMATION FOR NONLINEAR TIME FRACTIONAL TWO-TERM MIXED SUB-DIFFUSION AND DIFFUSION WAVE EQUATIONS

doi:10.4208/jcm.2104-m2020-0332

Journal of Computational Mathematics ›› 2022, Vol. 40 ›› Issue (6): 936-954.DOI: 10.4208/jcm.2104-m2020-0332

Previous Articles Next Articles

A TWO-GRID FINITE ELEMENT APPROXIMATION FOR NONLINEAR TIME FRACTIONAL TWO-TERM MIXED SUB-DIFFUSION AND DIFFUSION WAVE EQUATIONS

Yanping Chen¹, Qiling Gu², Qingfeng Li², Yunqing Huang²

1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
2. School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411199, China

Received:2020-12-25 Revised:2021-01-30 Online:2022-11-15 Published:2022-11-18
Contact: Yanping Chen, Email: yanpingchen@scnu.edu.cn
Supported by:
This work is supported by the State Key Program of National Natural Science Foundation of China (11931003) and National Natural Science Foundation of China (41974133, 11971410), Project for Hunan National Applied Mathematics Center of Hunan Provincial Science and Technology Department (2020ZYT003), Hunan Provincial Innovation Foundation for Postgraduate, China (XDCX2020B082, XDCX2021B098), Postgraduate Scientific Research Innovation Project of Hunan Province (CX20210597).

PDF ( 2 ) Cited

Yanping Chen, Qiling Gu, Qingfeng Li, Yunqing Huang. A TWO-GRID FINITE ELEMENT APPROXIMATION FOR NONLINEAR TIME FRACTIONAL TWO-TERM MIXED SUB-DIFFUSION AND DIFFUSION WAVE EQUATIONS[J]. Journal of Computational Mathematics, 2022, 40(6): 936-954.

In this paper, we develop a two-grid method (TGM) based on the FEM for 2D nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations. A two-grid algorithm is proposed for solving the nonlinear system, which consists of two steps: a nonlinear FE system is solved on a coarse grid, then the linearized FE system is solved on the fine grid by Newton iteration based on the coarse solution. The fully discrete numerical approximation is analyzed, where the Galerkin finite element method for the space derivatives and the finite difference scheme for the time Caputo derivative with order α ∈ (1, 2) and α₁ ∈ (0, 1). Numerical stability and optimal error estimate O(h^r+1 + H^2r+2 + τ^{min-3-α,2-α1}}) in L²-norm are presented for two-grid scheme, where t, H and h are the time step size, coarse grid mesh size and fine grid mesh size, respectively. Finally, numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.

Key words: Two-grid method, Finite element method, Nonlinear time fractional mixed sub-diffusion and diffusion-wave equations, L1-CN scheme, Stability and convergence

CLC Number:

[1] L. Chen and Y. Chen, Two-grid method for nonlinear reaction-diffusion equations by mixed finite element methods, J. Sci. Comput., 49(2011), 383-401.
[2] Y. Chen, Y. Huang, and D. Yu, A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations, Int. J. Numer. Meth. Engng., 57(2003), 193-209.
[3] P. Ciarlet, The finite element method for elliptic problems, North-Holland, New York, 1978.
[4] S. Das and I. Dan, Fractional Order Signal Processing: Introductory Concepts and Applications, Springer, 2011.
[5] L. Feng, F. Liu, and Ian Turner, Novel numerical analysis of multi-term time fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD Couette flow of a generalized OldroydB fluid, Fract. Calc. Appl. Anal., 21(2018), 1073-1103.
[6] B.I. Henry and S.L. Wearne, Fractional reaction-diffusion, Phys. A, 276(2000), 448-455.
[7] R. Herrmann, Fractional Calculus: An Introduction for Physicists, World Scientific, 2011.
[8] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 1999.
[9] B. Jin, L. Raytcho, and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA Journal of Numerical Analysis., 36(2016), 197-221.
[10] R.C. Koeller, Application of fractional calculus to the theory of viscoelasticity, J. Appl. Mech., 51(1984), 229-307.
[11] Q. Li, Y. Chen, Y. Huang, and Y. Wang, Two-grid methods for nonlinear time fractional diffusion equations by L1-Galerkin FEM, Mathematics and Computers in Simulation., 185(2021), 436-451.
[12] Q. Li, Y. Chen, Y. Huang, and Y. Wang. Two-grid methods for semilinear time fractional reaction diffusion equations by expanded mixed finite element method, Appl. Numer. Math., 157(2020), 38-54.
[13] B. Li, T. Wang, and X. Xie, Analysis of the L1 scheme for fractional wave equations with nonsmooth data, Comput.Math.Appl., 90(2021), 1-12.
[14] F. Liu, P. Zhuang, and Q. Liu, Numerical Methods and Applications for Fractional Partial Differential Equations, Science Press, Beijing, 2015.
[15] Y. Liu, Y. Du, H. Li, and J. Wang, A two-grid finite element approximation for a nonlinear time-fractional Cable equation, Nonlinear Dyn., 85(2016), 2535-2548.
[16] Y. Liu, Y. Du, H. Li, J. Li, and S. He, A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative, Comput. Math. Appl., 70(2015), 2474-2492.
[17] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer, Berlin, 1997.
[18] J. Ren and Z. Sun, Numerical algorithm with high spatial accuracy for the fractional diffusion-wave equation with neumann boundary conditions, J. Sci. Comput., 56(2013), 381-408.
[19] M. Stiassnie, On the application of fractional calculus for the formulation of viscoelastic models, Appl. Math. Model., 3(1979), 300-302.
[20] Z. Sun, C. Ji, and R. Du, A new analytical technique of the L-type difference schemes for time fractional mixed sub-diffusion and diffusion-wave equations, Appl. Math. Lett., 102(2020), 106-115.
[21] Z. Sun, The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations, Science Press, Beijing, 2009.
[22] Y. Wei, Y. Zhao, Y. Tang, F. Wang, Z. Shi, and K. Li, High accuracy analysis of FEM for two-term time-fractional mixed diffusion-wave equations(in Chinese), Sci. Sin. Inform., 48(2018), 871-887.
[23] J. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput., 15(1994), 231-237.
[24] J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM. J. Numer. Anal., 33(1996), 1759-1777.
[25] B. Yuste and K. Lindenberg, Subdiffusion-limited A + A reactions, Phys. Rev. Lett., 87(2001), doi: 10.1103/PhysRevLett.87.118301.
[26] Y. Zhang, Z. Sun, and X. Zhao, Compact alternating direction implicit scheme for the twodimensional fractional diffusion-wave equation, SIAM J. Numer. Anal., 50(2012), 1535-1555.
[27] Y. Zhao, F. Wang, X. Hu, Z. Shi, and Y. Tang, Anisotropic linear triangle finite element approximation for multi-term time-fractional mixed diffusion and diffusion-wave equations with variable coefficient on 2D bounded domain, Comput. Math. Appl., 78(2019), 1705-1719.
[28] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.

[1]	Baiying Dong, Xiufeng Feng, Zhilin Li. AN L^∞ SECOND ORDER CARTESIAN METHOD FOR 3D ANISOTROPIC INTERFACE PROBLEMS [J]. Journal of Computational Mathematics, 2022, 40(6): 882-912.
[2]	Bei Zhang, Jikun Zhao, Minghao Li, Hongru Chen. STABILIZED NONCONFORMING MIXED FINITE ELEMENT METHOD FOR LINEAR ELASTICITY ON RECTANGULAR OR CUBIC MESHES [J]. Journal of Computational Mathematics, 2022, 40(6): 865-881.
[3]	Tianliang Hou, Chunmei Liu, Chunlei Dai, Luoping Chen, Yin Yang. TWO-GRID ALGORITHM OF H¹-GALERKIN MIXED FINITE ELEMENT METHODS FOR SEMILINEAR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS [J]. Journal of Computational Mathematics, 2022, 40(5): 667-685.
[4]	Kai Wang, Na Wang. ANALYSIS OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR PARABOLIC INTERFACE PROBLEMS WITH NONSMOOTH INITIAL DATA [J]. Journal of Computational Mathematics, 2022, 40(5): 777-793.
[5]	Hanzhang Hu, Yanping Chen. A CHARACTERISTIC MIXED FINITE ELEMENT TWO-GRID METHOD FOR COMPRESSIBLE MISCIBLE DISPLACEMENT PROBLEM [J]. Journal of Computational Mathematics, 2022, 40(5): 794-813.
[6]	Abdelhamid Zaghdani, Sayed Sayari, Miled EL Hajji. A NEW HYBRIDIZED MIXED WEAK GALERKIN METHOD FOR SECOND-ORDER ELLIPTIC PROBLEMS* [J]. Journal of Computational Mathematics, 2022, 40(4): 499-516.
[7]	Xinjiang Chen, Yanqiu Wang. A CONFORMING QUADRATIC POLYGONAL ELEMENT AND ITS APPLICATION TO STOKES EQUATIONS [J]. Journal of Computational Mathematics, 2022, 40(4): 624-648.
[8]	Xiaonian Long, Qianqian Ding. A SECOND ORDER UNCONDITIONALLY CONVERGENT FINITE ELEMENT METHOD FOR THE THERMAL EQUATION WITH JOULE HEATING PROBLEM [J]. Journal of Computational Mathematics, 2022, 40(3): 354-372.
[9]	Ram Manohar, Rajen Kumar Sinha. ELLIPTIC RECONSTRUCTION AND A POSTERIORI ERROR ESTIMATES FOR FULLY DISCRETE SEMILINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS [J]. Journal of Computational Mathematics, 2022, 40(2): 147-176.
[10]	Weijie Huang, Wei Jiang, Yan Wang. A θ-L APPROACH FOR SOLVING SOLID-STATE DEWETTING PROBLEMS [J]. Journal of Computational Mathematics, 2022, 40(2): 275-293.
[11]	Kaibo Hu, Ragnar Winther. WELL-CONDITIONED FRAMES FOR HIGH ORDER FINITE ELEMENT METHODS [J]. Journal of Computational Mathematics, 2021, 39(3): 333-357.
[12]	Xiaodi Zhang, Weiying Zheng. MONOLITHIC MULTIGRID FOR REDUCED MAGNETOHYDRODYNAMIC EQUATIONS [J]. Journal of Computational Mathematics, 2021, 39(3): 453-470.
[13]	Xiaoliang Song, Bo Chen, Bo Yu. ERROR ESTIMATES FOR SPARSE OPTIMAL CONTROL PROBLEMS BY PIECEWISE LINEAR FINITE ELEMENT APPROXIMATION [J]. Journal of Computational Mathematics, 2021, 39(3): 471-492.
[14]	Xiaocui Li, Xu You. MIXED FINITE ELEMENT METHODS FOR FRACTIONAL NAVIER-STOKES EQUATIONS [J]. Journal of Computational Mathematics, 2021, 39(1): 130-146.
[15]	Michael Holst, Yuwen Li, Adam Mihalik, Ryan Szypowski. CONVERGENCE AND OPTIMALITY OF ADAPTIVE MIXED METHODS FOR POISSON'S EQUATION IN THE FEEC FRAMEWORK [J]. Journal of Computational Mathematics, 2020, 38(5): 748-767.

Viewed

Full text

Abstract

A TWO-GRID FINITE ELEMENT APPROXIMATION FOR NONLINEAR TIME FRACTIONAL TWO-TERM MIXED SUB-DIFFUSION AND DIFFUSION WAVE EQUATIONS

扫码分享