Analysis of a new stabilized finite volume element method based on multiscale enrichment for the Navier-Stokes problem

2016 ◽  
Vol 26 (8) ◽  
pp. 2462-2485 ◽  
Author(s):  
Juan Wen ◽  
Yinnian He ◽  
Xin Zhao
Keyword(s):  
Finite Volume ◽  
Stokes Problem ◽  
Volume Element ◽  
New Method ◽  
Navier Stokes ◽  
Optimal Order ◽  
Finite Volume Element Method ◽  
Content Type ◽  
Finite Volume Element ◽  
Element Method

Purpose The purpose of this paper is to propose a new stabilized finite volume element method for the Navier-Stokes problem. Design/methodology/approach This new method is based on the multiscale enrichment and uses the lowest equal order finite element pairs P1/P1. Findings The stability and convergence of the optimal order in H1-norm for velocity and L2-norm for pressure are obtained. Originality/value Using a dual problem for the Navier-Stokes problem, the convergence of the optimal order in L2-norm for the velocity is obtained. Finally, numerical example confirms the theory analysis and validates the effectiveness of this new method.

A MODIFIED IMMERSED FINITE VOLUME ELEMENT METHOD FOR ELLIPTIC INTERFACE PROBLEMS

2020 ◽  
Vol 62 (1) ◽  
pp. 42-61
Author(s):  
Q. WANG ◽  
Z. ZHANG
Keyword(s):  
Finite Volume ◽  
Control Volume ◽  
Second Order ◽  
Volume Element ◽  
New Method ◽  
Optimal Error Estimates ◽  
Local Conservation ◽  
Finite Volume Element Method ◽  
Finite Volume Element ◽  
Element Method

This paper presents a new immersed finite volume element method for solving second-order elliptic problems with discontinuous diffusion coefficient on a Cartesian mesh. The new method possesses the local conservation property of classic finite volume element method, and it can overcome the oscillating behaviour of the classic immersed finite volume element method. The idea of this method is to reconstruct the control volume according to the interface, which makes it easy to implement. Optimal error estimates can be derived with respect to an energy norm under piecewise $H^{2}$ regularity. Numerical results show that the new method significantly outperforms the classic immersed finite volume element method, and has second-order convergence in $L^{\infty }$ norm.

scholarly journals Discontinuous Galerkin Immersed Finite Volume Element Method for Anisotropic Flow Models in Porous Medium

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Zhong-yan Liu ◽  
Huan-zhen Chen
Keyword(s):  
Function Space ◽  
Discontinuous Galerkin ◽  
Finite Volume ◽  
Volume Element ◽  
Piecewise Constant Function ◽  
Optimal Order ◽  
Finite Volume Element Method ◽  
Norm Estimate ◽  
Finite Volume Element ◽  
Element Method

By choosing the trial function space to the immersed finite element space and the test function space to be piecewise constant function space, we develop a discontinuous Galerkin immersed finite volume element method to solve numerically a kind of anisotropic diffusion models governed by the elliptic interface problems with discontinuous tensor-conductivity. The existence and uniqueness of the discrete scheme are proved, and an optimal-order energy-norm estimate andL2-norm estimate for the numerical solution are derived.

A modified immersed finite volume element method for elliptic interface problems

2020 ◽  
Vol 62 ◽  
pp. 42-61 ◽  
Author(s):  
Quanxiang Wang ◽  
Zhiyue Zhang
Keyword(s):  
Finite Volume ◽  
Control Volume ◽  
Second Order ◽  
Volume Element ◽  
New Method ◽  
Optimal Error Estimates ◽  
Local Conservation ◽  
Finite Volume Element Method ◽  
Finite Volume Element ◽  
Element Method

This paper presents a new immersed finite volume element method for solving second-order elliptic problems with discontinuous diffusion coefficient on a Cartesian mesh. The new method possesses the local conservation property of classic finite volume element method, and it can overcome the oscillating behaviour of the classic immersed finite volume element method. The idea of this method is to reconstruct the control volume according to the interface, which makes it easy to implement. Optimal error estimates can be derived with respect to an energy norm under piecewise \(H^{2}\) regularity. Numerical results show that the new method significantly outperforms the classic immersed finite volume element method, and has second-order convergence in \(L^{\infty}\) norm. doi: 10.1017/S1446181120000073

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