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Numerical Analysis of Fourier Finite Volume Element Method for Dirichlet Boundary Optimal Control Problems Governed by Elliptic PDEs on Complex Connected Domains

Author

Listed:
  • Mengya Su

    (School of Mathematical Sciences, Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing 210023, China)

  • Liuqing Xie

    (School of Mathematical Sciences, Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing 210023, China)

  • Zhiyue Zhang

    (School of Mathematical Sciences, Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing 210023, China)

Abstract

In this research, we investigate an optimal control problem governed by elliptic PDEs with Dirichlet boundary conditions on complex connected domains, which can be utilized to model the cooling process of concrete dam pouring. A new convergence result for two-dimensional Dirichlet boundary control is proven with the Fourier finite volume element method. The Lagrange multiplier approach is employed to find the optimality systems of the Dirichlet boundary optimal control problem. The discrete optimal control problem is then obtained by applying the Fourier finite volume element method based on Galerkin variational formulation for optimality systems, that is, using Fourier expansion in the azimuthal direction and the finite volume element method in the radial direction, respectively. In this way, the original two-dimensional problem is reduced to a sequence of one-dimensional problems, with the Dirichlet boundary acting as an interval endpoint at which a quadratic interpolation scheme can be implemented. The convergence order of state, adjoint state, and Dirichlet boundary control are therefore proved. The effectiveness of the method is demonstrated numerically, and numerical data is provided to support the theoretical analysis.

Suggested Citation

  • Mengya Su & Liuqing Xie & Zhiyue Zhang, 2022. "Numerical Analysis of Fourier Finite Volume Element Method for Dirichlet Boundary Optimal Control Problems Governed by Elliptic PDEs on Complex Connected Domains," Mathematics, MDPI, vol. 10(24), pages 1-26, December.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:24:p:4779-:d:1004746
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    References listed on IDEAS

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    1. Ewing, Richard E. & Li, Zhilin & Lin, Tao & Lin, Yanping, 1999. "The immersed finite volume element methods for the elliptic interface problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 50(1), pages 63-76.
    2. Tuğba Akman & Hamdullah Yücel & Bülent Karasözen, 2014. "A priori error analysis of the upwind symmetric interior penalty Galerkin (SIPG) method for the optimal control problems governed by unsteady convection diffusion equations," Computational Optimization and Applications, Springer, vol. 57(3), pages 703-729, April.
    3. Eduardo Casas & Vili Dhamo, 2012. "Error estimates for the numerical approximation of Neumann control problems governed by a class of quasilinear elliptic equations," Computational Optimization and Applications, Springer, vol. 52(3), pages 719-756, July.
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