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Time-stepping discontinuous Galerkin approximation of optimal control problem governed by time fractional diffusion equation

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Abstract

In this paper, a piecewise constant time-stepping discontinuous Galerkin method combined with a piecewise linear finite element method is applied to solve control constrained optimal control problem governed by time fractional diffusion equation. The control variable is approximated by variational discretization approach. The discrete first-order optimality condition is derived based on the first discretize then optimize approach. We demonstrate the commutativity of discretization and optimization for the time-stepping discontinuous Galerkin discretization. Since the state variable and the adjoint state variable in general have weak singularity near t = 0and t = T, a time adaptive algorithm is developed based on step doubling technique, which can be used to guide the time mesh refinement. Numerical examples are given to illustrate the theoretical findings.

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Acknowledgements

The author would like to thank the support of National Natural Science Foundation of China (No. 11301311 and 11471196) and Natural Science Foundation of Shandong Province (No. ZR2016JL004).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, 250014, Ji’nan, China

    Zhaojie Zhou & Chenyang Zhang

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  1. Zhaojie Zhou
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  2. Chenyang Zhang
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Correspondence to Zhaojie Zhou.

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Zhou, Z., Zhang, C. Time-stepping discontinuous Galerkin approximation of optimal control problem governed by time fractional diffusion equation. Numer Algor 79, 437–455 (2018). https://doi.org/10.1007/s11075-017-0445-3

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