A Uzawa-Type Iterative Algorithm for the Stationary Natural Convection Model

Area of Science:

• Computational fluid dynamics
• Numerical analysis
• Partial differential equations

Background:

• Stationary natural convection models are crucial in fluid dynamics and heat transfer.
• Existing iterative algorithms may not guarantee essential physical properties like divergence-free velocity.
• Mixed finite element methods are widely used for discretizing fluid flow equations.

Purpose of the Study:

• To introduce and analyze a novel Uzawa-type iterative algorithm.
• To address the challenge of achieving weakly divergence-free velocity approximations.
• To solve the stationary natural convection model using mixed finite element discretization.

Main Methods:

• Development of a modified Uzawa-type iterative algorithm.
• Application of mixed finite element methods for spatial discretization.
• Theoretical analysis of algorithm convergence properties.

Main Results:

• The proposed algorithm yields a weakly divergence-free velocity approximation.
• Demonstrated improvement over common Uzawa iterative algorithms.
• Convergence of the new algorithm is theoretically established.

Conclusions:

• The novel Uzawa-type algorithm effectively solves stationary natural convection problems.
• The method provides a physically meaningful weakly divergence-free velocity field.
• Numerical results validate the theoretical convergence findings.