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Penalty Virtual Element Method for the 3D Incompressible Flow on Polyhedron Mesh
Lulu Li1, Haiyan Su1, Yinnian He1
1College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China.
Entropy (Basel, Switzerland)
|August 26, 2022
Summary
A new penalty virtual element method (VEM) effectively solves 3D incompressible flow on polyhedral meshes. This method bypasses complex computations, offering a robust solution for fluid dynamics simulations.
Area of Science:
- Computational Fluid Dynamics
- Numerical Analysis
- Applied Mathematics
Background:
- Standard finite element methods face challenges with arbitrary meshes.
- Virtual Element Method (VEM) offers a mesh-independent approach.
- Solving 3D incompressible flow requires robust numerical techniques.
Purpose of the Study:
- To propose and analyze a penalty virtual element method (VEM) for 3D incompressible flow.
- To address limitations of standard finite element discretizations on complex geometries.
- To provide a rigorous error analysis for the proposed method.
Main Methods:
- Implementation of a penalty virtual element method (VEM) on polyhedral meshes.
- Approximation of velocity and pressure using lowest equal-order virtual element spaces (Xh, Qh).
- Application of a penalty method to handle the inf-sup condition deficiency.
Main Results:
- Rigorous error estimation was proven for the VEM approach.
- The method successfully handles 3D incompressible flow problems.
- Numerical results validated the theoretical findings and method's effectiveness.
Conclusions:
- The penalty VEM is a viable and effective method for 3D incompressible flow.
- VEM simplifies computations by avoiding explicit trial and test space calculations.
- The method demonstrates strong performance on 3D polygonal meshes.
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