We propose high order conforming and nonconforming immersed hybridized difference (IHD) methods in two and three dimensions for elliptic interface problems. Introducing the virtual to real transformation (VRT), we could obtain a systematic and unique way of deriving arbitrary high order methods in principle. The optimal number of collocating points for imposing interface conditions is proved, and a unique way of constructing the VRT is suggested. Numerical experiments are performed in two and three dimensions. Numerical results achieving up to the 6th order convergence in the L 2-norm are presented for the two dimensional case, and a three dimensional example with a 4th order convergence is presented.

中文翻译：

---

椭圆界面问题的高阶浸入式杂化差分法

我们提出了二维和三维的高阶一致和非一致浸入式杂化差分（IHD）方法来解决椭圆界面问题。引入虚实变换（VRT），原则上我们可以获得一种系统的、独特的推导任意高阶方法的方法。证明了施加界面条件的最佳配置点数量，并提出了构建VRT的独特方法。数值实验在二维和三维上进行。数值结果达到 6 阶收敛L 2给出了二维情况的-范数，并给出了具有四阶收敛的三维示例。