Abstract
In this work we present numerical analysis for a distributed optimal control problem, with box constraint on the control, governed by a subdiffusion equation that involves a fractional derivative of order $alpha in (0,1)$ in time. The fully discrete scheme is obtained by applying the conforming linear Galerkin finite element method in space, L1 scheme/backward Euler convolution quadrature in time, and the control variable by a variational-type discretization. With a space mesh size $h$ and time stepsize $tau $ we establish the following order of convergence for the numerical solutions of the optimal control problem: $O(tau {min ({1}/{2}+alpha -epsilon, 1)}+h2)$ in the discrete $L2(0,T;L2(varOmega)) $ norm and $O(tau {5alpha 5656epsilon }.3.+ell56 h562h52)$ in the discrete $L{infty }(0,T;L2(varOmega)) $ norm, with any small $epsilon>0$ and $ell h=ln (2+1/h)$. The analysis relies essentially on the maximal $Lp$-regularity and its discrete analogue for the subdiffusion problem. Numerical experiments are provided to support the theoretical results.
Original language | English |
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Pages (from-to) | 377-404 |
Number of pages | 28 |
Journal | IMA Journal of Numerical Analysis |
Volume | 40 |
Issue number | 1 |
DOIs | |
Publication status | Published - Jan 2020 |
Keywords
- convolution quadrature
- L1 scheme
- maximal regularity
- optimal control
- pointwise-in-time error estimate
- time-fractional diffusion
ASJC Scopus subject areas
- General Mathematics
- Computational Mathematics
- Applied Mathematics