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Taiwo, O.A., Alhassan, L.K., Odetunde, O.S. and Alabi, O.O. (2023) Galerkin Method for Numerical Solution of Volterra Integro-Differential Equations with Certain Orthogonal Basis Function. International Journal of Modern Nonlinear Theory and Application, 12, 68-80.
https://doi.org/10.4236/ijmnta.2023.122005

has been cited by the following article:

• TITLE: Gradient Recovery Based Two-Grid Finite Element Method for Parabolic Integro-Differential Optimal Control Problems

  AUTHORS: Miao Yang

  KEYWORDS: Optimal Control Problem, Gradient Recovery, Two-Grid Finite Element Method

  JOURNAL NAME: Journal of Applied Mathematics and Physics, Vol.12 No.8, August 19, 2024

  ABSTRACT: In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and co-state variables, and piecewise constant function is used to approximate control variables. Generally, the optimal conditions for the problem are solved iteratively until the control variable reaches error tolerance. In order to calculate all the variables individually and parallelly, we introduce a gradient recovery based two-grid method. First, we solve the small scaled optimal control problem on coarse grids. Next, we use the gradient recovery technique to recover the gradients of state and co-state variables. Finally, using the recovered variables, we solve the large scaled optimal control problem for all variables independently. Moreover, we estimate priori error for the proposed scheme, and use an example to validate the theoretical results.