Nonconforming Immersed Finite Element Spaces For Elliptic Interface Problems
It provides theoretical foundations for nonconforming IFE methods, which are important for solving interface problems in computational science and engineering.
This paper proves the existence, uniqueness, and optimal approximation capability of two classes of nonconforming immersed finite element spaces for elliptic interface problems, using a unified framework and Sherman-Morrison systems.
In this paper, we use a unified framework introduced in [3] to study two classes of 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 shape functions on each interface element in either 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 that these IFE spaces have the optimal approximation capability.