Convergence and Superconvergence of Fully Discrete Finite Element for Time Fractional Optimal Control Problems

Yuelong Tang

College of Science, Hunan University of Science and Engineering, Yongzhou, China.

DOI: 10.4236/ajcm.2021.111005                

Abstract

In this paper, we consider a fully discrete finite element approximation for time fractional optimal control problems. The state and adjoint state are approximated by triangular linear finite elements in space and L1 scheme in time. The control is obtained by the variational discretization technique. The main purpose of this work is to derive the convergence and superconvergence. A numerical example is presented to validate our theoretical results.

Keywords

Time Fractional Optimal Control Problems, Finite Element, Convergence and Superconvergence

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

[1]	Ciarlet, P. (1978) The Finite Element Method for Elliptic Problems. Amstterdam, North-Holland.
[2]	Liu, W. and Yan, N. (2008) Adaptive Finite Element Methods for Optimal Control Governed by PDEs. Science Press, Beijing.
[3]	Hinze, M., Pinnau, R., Ulbrich, M. and Ulbrich S. (2009) Optimization with PDE Constraints. Springer, New York.
[4]	Gao, G. and Sun, Z. (2011) A Compact Finite Difference Scheme for the Fractional Sub-Diffusion Equations. Journal of Computational Physics, 230, 586-595. https://doi.org/10.1016/j.jcp.2010.10.007
[5]	Lin, Y. and Xu, C. (2007) Finite Difference/Spectral Approximation for the Time-Fractional Diffusion Equation. Journal of Computational Physics, 225, 1533-1552. https://doi.org/10.1016/j.jcp.2007.02.001
[6]	Li, X. and Xu, C. (2009) A Space-Time Spectral Method for the Time Fractional Diffusion Equation. SIAM Journal on Numerical Analysis, 47, 2108-2131. https://doi.org/10.1137/080718942
[7]	Mao, Z. and Shen, J. (2017) Hermite Spectral Methods for Fractional PDEs in Unbounded Domains. SIAM Journal on Scientific Computing, 39, A1928-A1950. https://doi.org/10.1137/16M1097109
[8]	Zheng, M., Liu, F., Anh, V. and Turner, I. (2016) A High-Order Spectral Method for the Multi-Term Time-Fractional Diffusion Equations. Applied Mathematical Modelling, 40, 4970-4985. https://doi.org/10.1016/j.apm.2015.12.011
[9]	Shi, Z., Zhao, Y., Liu, F., Tang, Y., et al. (2017) High Accuracy Analysis of an H1-Galerkin Mixed Finite Element Method for Two-Dimensional Time Fractional Diffusion Equations. Computers and Mathematics with Applications, 74, 1903-1914. https://doi.org/10.1016/j.camwa.2017.06.057
[10]	Zhao, Y., Chen, P., Bu, W., et al. (2017) Two Mixed Finite Element Methods for Time-Fractional Diffusion Equations. Journal of Scientific Computing, 70, 407-428. https://doi.org/10.1007/s10915-015-0152-y
[11]	Chen, L., Nochetto, R., Otárola, E. and Salgado, A. (2016) Multilevel Methods for Nonuniformly Elliptic Operators and Fractional Diffusion. Mathematics of Computation, 85, 2583-2607. https://doi.org/10.1090/mcom/3089
[12]	Jin, B., Lazarov, R., Pasciak, J. and Zhou, Z. (2015) Error Analysis of Semidiscrete Finite Element Methods for Inhomogeneous Time-Fractional Diffusion. IMA Journal of Numerical Analysis, 35, 561-582. https://doi.org/10.1093/imanum/dru018
[13]	Zeng, F., Li, C., Liu, F. and Turner, I. (2015) Numerical Alogrithms for Time Fractional Subdiffusion Equation with Second-Order Accuracy. SIAM Journal on Scientific Computing, 37, A55-A78. https://doi.org/10.1137/14096390X
[14]	Zhou, Z. and Gong, W. (2016) Finite Element Approximation of Optimal Control Problems Governed by Time Fractional Diffusion Equation. Computers and Mathematics with Applications, 71, 301-318. https://doi.org/10.1016/j.camwa.2015.11.014
[15]	Du, N., Wang, H. and Liu, W. (2016) A Fast Gradient Projection Method for a Constrained Fractional Optimal Control. Journal of Scientific Computing, 68, 1-20. https://doi.org/10.1007/s10915-015-0125-1
[16]	Zhou, Z. and Tan, Z. (2019) Finite Element Approximation of Optimal Control Problem Governed by Space Fractional Equation. Journal of Scientific Computing, 78, 1840-1861. https://doi.org/10.1007/s10915-018-0829-0
[17]	Zhang, C., Liu, H. and Zhou, Z. (2019) A Priori Error Analysis for Time-Stepping Discontinuous Galerkin Finite Element Approximation of Time Fractional Optimal Control Problem. Journal of Scientific Computing, 80, 993-1018. https://doi.org/10.1007/s10915-019-00964-9
[18]	Gunzburger, M. and Wang, J. (2019) Error Analysis of Fully Discrete Finite Element Approximations to an Optimal Control Problem Governed by a Time-Fractional PDE. SIAM Journal on Control and Optimization, 57, 241-263. https://doi.org/10.1137/17M1155636
[19]	Lions, J. and Magenes, E. (1972) Non Homogeneous Boundary Value Problems and Applications. Springer-Verlag, Berlin.
[20]	Hinze, M., Yan, N. and Zhou, Z. (2009) Variational Discretization for Optimal Control Governed by Convection Dominated Diffusion Equations. Journal of Computational Mathematics, 27, 237-253. http://global-sci.org/intro/article detail/jcm/8570.html
[21]	Tang, Y. and Hua, Y. (2014) Supercovergence Analysis for Parabolic Optimal Control Problems. Calcolo, 51, 381-392. https://doi.org/10.1007/s10092-013-0091-7
[22]	Chen, Y., Huang, Y. and Yi, N. (2008) A Posteriori Error Estimates of Spectral Method for Optimal Control Problems Governed by Parabolic Equations. Science in China Series A: Mathematics, 51, 1376-1390. https://doi.org/10.1007/s11425-008-0097-9
[23]	Jiang, Y. and Ma, J. (2011) High-Order Finite Element Methods for Time-Fractional Partial Di_erential Equations. Journal of Computational and Applied Mathematics, 235, 3285-3290. https://doi.org/10.1016/j.cam.2011.01.011
[24]	Li, R., Liu, W. and Yan, N. (2007) A Posteriori Error Estimates of Recovery Type for Distributed Convex Optimal Control Problems. Journal of Scientific Computing, 33, 155-182. https://doi.org/10.1007/s10915-007-9147-7