Quadratic Immersed Finite Element Method For Second-order Elliptic Interface Problems

We present quadratic immersed finite element (IFE) spaces that can be used to be solve two dimensional second order elliptic interface problems. Conventional finite ele-ment method can solve the problem with homogeneous material, such as the differential equation in continuum mechanics, the Maxwell equation in electromagnetic field and so on. However this methods have to be tailored such that their meshes are with the ma-terial interface for a non uniform material, such as electromagnetic problems, charging in space problems. This method spend a large amount of computation. The immersed finite element method mentioned in this paper is nonconforming and its partition can be independent of the interface, so we can choose the fixed grid, such as Cartesian meshes.This saves considerable amount of time and storage.Generally, the basis functions of the finite element methods for interface problem are made to satisfy the freedom condition and jump conditions, but to construct piece-wise quadratic IFE shape functions, extra constraints need to carefully introduced, and the existence and uniqueness are proofed. In our paper ,we first introduce two types of the extended interface jump conditions for linear interface,after that, we extend the framework to quadratic interface. The interpolation errors in the proposed piecewise 2th degree spaces yield optimal ?(h3) and ?(h2) convergence rates in the L2 and broken Hl norms,respectively,under mesh refinement. Furthermore, numerical experiments are pre-sented. However, the immersed finite methods formulation does not converge optimally neither in the L2 space or the broken H1 space. Based on those reasons, a partially penalized method combined with the proposed quadratic IFE spaces is developed which also converges optimally . While this penalty is not needed when either linear or bilinear IFE space is used.