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A Priori Error Analysis for Time-Stepping Discontinuous Galerkin Finite Element Approximation of Time Fractional Optimal Control Problem

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Abstract

In this paper a priori error analysis for time-stepping discontinuous Galerkin finite element approximation of optimal control problem governed by time fractional diffusion equation is presented. A time-stepping discontinuous Galerkin finite element method and variational discretization approach are used to approximate the state and control variables respectively. Regularity of the optimal control problem is discussed. Since the time fractional derivative is nonlocal, in order to reduce the computational cost a fast gradient projection algorithm is designed for the control problem based on the block triangular Toeplitz structure of the discretized state equation and adjoint state equation. Numerical examples are carried out to illustrate the theoretical findings and fast algorithm.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, China

    Chenyang Zhang & Zhaojie Zhou

  2. Institute of Applied Physics and Computational Mathematics, Beijing, China

    Huipo Liu

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  1. Chenyang Zhang
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  2. Huipo Liu
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  3. Zhaojie Zhou
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Correspondence to Zhaojie Zhou.

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The first and third author would like to thank the support of Natural Science Foundation of Shandong Province (No. ZR2016JL004) and National Natural Science Foundation of China (No. 11301311). The second author is supported by National Natural Science Foundation of China (No. 11771053).

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Zhang, C., Liu, H. & Zhou, Z. A Priori Error Analysis for Time-Stepping Discontinuous Galerkin Finite Element Approximation of Time Fractional Optimal Control Problem. J Sci Comput 80, 993–1018 (2019). https://doi.org/10.1007/s10915-019-00964-9

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  • DOI: https://doi.org/10.1007/s10915-019-00964-9

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