We present a unified framework for developing and analysing immersed finite
element (IFE) spaces for solving typical elliptic interface problems with
interface independent meshes. This framework allows us to construct a group of
new IFE spaces with either linear, or bilinear, or the rotated-Q1 polynomials.
Functions in these IFE spaces are locally piecewise polynomials defined
according to the sub-elements formed by the interface itself instead of its
line approximation. We show that the unisolvence for these IFE spaces follows
from the invertibility of the Sherman-Morrison matrix. A group of estimates and
identities are established for the interface geometry and shape functions that
are applicable to all of these IFE spaces. Most importantly, these fundamental
preparations enable us to develop a unified multipoint Taylor expansion
procedure for proving that these IFE spaces have the expected optimal
approximation capability according to the involved polynomials.

我们提出了一个统一的框架，用于开发和分析浸入式有限元 (IFE) 空间，以便解决具有界面无关网格的典型椭圆界面问题。该框架允许我们构建一组新的 IFE 空间，其中包括线性的、双线性的或旋转 Q1 的多项式。在这些 IFE 空间中的函数是局部分段多项式，其定义取决于由界面本身而不是其线性逼近形成的子元素。我们证明了这些 IFE 空间的单值性来自 Sherman-Morrison 矩阵的可逆性。还建立了适用于所有这些 IFE 空间的界面几何和形状函数的一组估计和恒等式。最重要的是，这些基本准备使我们能够开发统一的多点 Taylor 扩展过程，以证明这些 IFE 空间具有所涉及多项式的期望最优逼近能力。