Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint

Bangti Jin, Buyang Li, Zhi Zhou

Research output: Journal article publicationJournal articleAcademic researchpeer-review

5 Citations (Scopus)

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 languageEnglish
Pages (from-to)377-404
Number of pages28
JournalIMA Journal of Numerical Analysis
Volume40
Issue number1
DOIs
Publication statusPublished - Jan 2020

Keywords

  • convolution quadrature
  • L1 scheme
  • maximal regularity
  • optimal control
  • pointwise-in-time error estimate
  • time-fractional diffusion

ASJC Scopus subject areas

  • Mathematics(all)
  • Computational Mathematics
  • Applied Mathematics

Cite this