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

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.

A Stabilized Finite Volume Element Method for Stationary Stokes–Darcy Equations Using the Lowest Order

2019 ◽  
Vol 17 (08) ◽  
pp. 1950053
Author(s):  
Yanyun Wu ◽  
Liquan Mei ◽  
Meilan Qiu ◽  
Yuchuan Chu
Keyword(s):  
Finite Volume ◽  
Mass Conservation ◽  
Volume Element ◽  
Optimal Error Estimates ◽  
Irregular Domain ◽  
Finite Volume Element Method ◽  
Stabilized Finite Element Method ◽  
Finite Volume Element ◽  
Text Element ◽  
Element Method

We present a stabilized finite volume element method for the coupled Stokes–Darcy problem with the lowest order [Formula: see text] element for the Stokes region and [Formula: see text] element for the Darcy region. Based on adding a jump term of discrete pressure to the approximation equation, a discrete inf-sup condition is established for the proposed method. The optimal error estimates in the [Formula: see text]-norm for the velocity and piezometric head and in the [Formula: see text]-norm for the pressure are proved. And they are also verified through some numerical experiments. Two figures are given to show the full comparison for the local mass conservation between the proposed method and the stabilized finite element method. And this method can also be computed directly in the irregular domain according to the last experiment.

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.

Sign in / Sign up

Export Citation Format

Share Document