A weak formulation for solving elliptic interface problems without body fitted grid

A typical elliptic interface problem is casted as piecewise defined elliptic partial differential equations (PDE) in different regions which are coupled together with interface conditions, such as jumps in solution and flux across the interface. In many situations, such as the interface is moving, the challenge is how to solve such a problem accurately, robustly and efficiently without generating a body fitted mesh. The key issue is how to capture complex geometry of the interface and jump conditions across the interface effectively on a fixed mesh while the interface is not aligned with the mesh and the PDE is not valid across the interface. In this work we present a systematic formulation and further study of a second order accurate numerical method proposed in Hou and Liu (2005) [16] for elliptic interface problem. The key idea is to decompose the solution into two parts, a singular part and a regular part. The singular part captures the interface conditions while the regular part belongs to an appropriate space in the whole domain, which can be solved by a standard finite element formulation. In a general setup the two parts are coupled together. We give an explicit study of the construction of the singular part and the discretized system for the regular part. One key advantage of using weak formulation is that one can avoid assuming/using more regularity than necessary of the solution and the interface. We present the numerical algorithm and numerical tests in 3D to demonstrate the accuracy and other properties of our method.