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.

中文翻译：

---

椭圆界面问题的一种改进的浸入有限体积元方法

本文提出了一种新的浸入式有限体积元方法，用于求解笛卡尔网格上具有不连续扩散系数的二阶椭圆问题。新方法具有经典有限体积元法的局部守恒特性，可以克服经典浸没式有限体积元法的振荡行为。该方法的思想是根据接口重构控制量，便于实现。可以根据分段下的能量范数得出最佳误差估计$H^{2}$规律性。数值结果表明，新方法明显优于经典的浸没有限体积元方法，并且具有二阶收敛性$L^{\infty }$规范。