Optimal Control of Stationary Doubly Diffusive Flows on Bounded Lipschitz Domains: Numerical Analysis

Concetti Chiave

Proposing nonconforming discretizations for optimal control of double diffusion models.

Sintesi

In this work, a study on the optimal control of stationary doubly diffusive flows is presented. The authors propose discretizations based on lowest order Crouziex-Raviart finite element and piecewise constant spaces. They discuss the well-posedness of discrete uncontrolled state and adjoint equations using lifting and fixed point arguments. Convergence results are rigorously derived under minimal regularity. The local optimality of a reference control is proven using second-order sufficient optimality conditions, along with error estimates for control, state, and adjoint variables. Computational tests validate error decay rates and demonstrate applicability to thermohaline circulation problems.

Statistiche

Funding supported by SERB-CRG India (Grant Number : CRG/2021/002569).
Authors from Monash University, Melbourne, Australia, and Indian Institute of Technology, Roorkee, India.
Key words: Doubly diffusive flows, optimal control, mixed finite element method.
AMS subject classifications: 65N30, 65N50, 74F99, 74A50, 76S05.

Citazioni

Approfondimenti chiave tratti da

Optimal Control of Stationary Doubly Diffusive Flows on Two and Three Dimensional Bounded Lipschitz Domains

by Jai Tushar,A... alle arxiv.org 03-18-2024

https://arxiv.org/pdf/2403.10282.pdf

Domande più approfondite

How does the proposed methodology compare to existing numerical methods for simulating double-diffusive equations

The proposed methodology in the study offers a unique approach to numerically analyze optimal control of stationary doubly diffusive flows. By utilizing fully nonconforming finite element discretizations based on lowest order Crouziex-Raviart finite elements and piecewise constant spaces, the researchers aim to ensure locally exact divergence-free solutions. This method sets itself apart from existing numerical techniques by focusing on rigorous convergence results under minimal regularity assumptions. The use of discrete lifting and fixed point arguments enhances the well-posedness of the discrete state and adjoint equations, providing a robust framework for studying optimal control in double diffusion models.

What are the implications of the assumptions made on boundary data regularity and viscosity continuity in practical applications

The assumptions made regarding boundary data regularity and viscosity continuity play crucial roles in practical applications of the findings. The requirement for boundary data regularity ensures that realistic scenarios are considered where specific conditions are imposed at domain boundaries, reflecting physical constraints or known values in real-world systems. On the other hand, assuming Lipschitz continuity and uniform boundedness of kinematic viscosity is essential for maintaining stability and accuracy in simulations. These assumptions provide a solid foundation for modeling thermohaline circulation problems accurately while allowing for tractable computational implementations.

How can the findings in this study be extended to more complex fluid flow systems beyond stationary doubly diffusive flows

The insights gained from this study can be extended to more complex fluid flow systems beyond stationary doubly diffusive flows by adapting the methodologies developed here to suit different contexts. For instance, similar optimization techniques could be applied to transient double diffusion models or multiphase flow problems with varying viscosities or densities. Additionally, incorporating additional physics such as chemical reactions or phase changes into the control framework could lead to novel applications in environmental remediation or industrial processes. By building upon the principles established in this research, future studies can explore diverse fluid dynamics scenarios with enhanced control strategies tailored to specific system requirements.