Nonconforming immersed finite element spaces for elliptic interface problems

In this paper, we use a unified framework introduced in Chen and Zou (1998) to study two nonconforming immersed finite element (IFE) spaces with integral-value degrees of freedom. The shape functions on interface elements are piecewise polynomials defined on sub-elements separated either by the actual interface or its line approximation. In this unified framework, we use the invertibility of the well known Sherman-Morison systems to prove the existence and uniqueness of IFE shape functions on each interface element in either a rectangular or triangular mesh. Furthermore, we develop a multi-edge expansion for piecewise functions and a group of identities for nonconforming IFE functions which enable us to show the optimal approximation capability of these IFE spaces.