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SUPG-Stabilized Virtual Element Method for Optimal Control Problem Governed by a Convection Dominated Diffusion

Qiming Wang1, Zhaojie Zhou1

  • 1School of Mathematics and Statistics, Shandong Normal University, Jinan 250000, China.

Entropy (Basel, Switzerland)
|July 2, 2021
PubMed
Summary
This summary is machine-generated.

This study introduces a stabilized virtual element method for convection-dominated optimal control problems. The novel approach provides accurate estimates for state, adjoint state, and control variables.

Keywords:
SUPG stabilized VEMa priori error estimateconvection dominated diffusion equationoptimal control

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Area of Science:

  • Numerical analysis
  • Computational mathematics
  • Scientific computing

Background:

  • Optimal control problems governed by convection-dominated diffusion equations present significant numerical challenges.
  • Existing methods may struggle with accuracy and stability in these regimes.
  • The virtual element method (VEM) offers a flexible framework for complex geometries.

Purpose of the Study:

  • To investigate the streamline upwind/Petrov Galerkin (SUPG) stabilized virtual element method (VEM) for optimal control problems.
  • To develop and analyze a discrete scheme based on a first-optimize-then-discretize strategy.
  • To derive a priori error estimates for the state, adjoint state, and control.

Main Methods:

  • Construction of a virtual element discrete scheme using the first-optimize-then-discretize strategy.
  • Application of SUPG stabilization to VEM for both the state and adjoint state equations.
  • Derivation of a priori error estimates for the key variables.

Main Results:

  • The proposed SUPG-stabilized VEM effectively handles optimal control problems with convection-dominated diffusion.
  • A priori error estimates were successfully derived for the state, adjoint state, and control.
  • Numerical experiments validated the theoretical findings and demonstrated the method's efficacy.

Conclusions:

  • The SUPG-stabilized VEM is a robust and accurate numerical method for convection-dominated optimal control.
  • The derived error estimates provide theoretical guarantees for the method's performance.
  • This work contributes to the advancement of numerical techniques for complex control problems.