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Finite Element Iterative Methods for the 3D Steady Navier--Stokes Equations.
1School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China.
Entropy (Basel, Switzerland)
|December 24, 2021
Summary
This study introduces a finite element (FE) method for solving 3D steady Navier-Stokes equations. The method ensures convergence of the FE solution to the exact solution using iterative techniques and specific FE spaces.
Area of Science:
- Computational Fluid Dynamics
- Numerical Analysis
- Partial Differential Equations
Background:
- The 3D steady Navier-Stokes equations govern fluid flow but are challenging to solve numerically.
- Finite element methods (FEM) are widely used for solving differential equations, requiring suitable element pairs for stability and accuracy.
Purpose of the Study:
- To develop and analyze a finite element (FE) method for the 3D steady Navier-Stokes equations.
- To establish the existence, uniqueness, and convergence properties of the FE solution.
Main Methods:
- Utilizing the finite element pair Xh×Mh satisfying the discrete inf-sup condition.
- Employing Stokes, Newton, and Oseen iterative methods to transmit solutions.
- Presenting weak formulations for the linearized Navier-Stokes equations.
Main Results:
- Demonstrated existence and uniqueness of the FE solution (uhn,phn) for iterative equations.
- Deduced convergence of the FE solution (uhn,phn) to the exact solution (u,p) in the H1-L2 norm.
- Provided the convergence order for the FE velocity uhn to the exact velocity u in the L2 norm.
Conclusions:
- The proposed FE method is a viable approach for solving 3D steady Navier-Stokes equations.
- The discrete inf-sup condition and iterative methods are crucial for the method's stability and convergence.
- The analysis provides theoretical guarantees for the accuracy of the FE solution.
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