Analysis of nonconforming IFE methods and a new scheme for elliptic interface problems

¹ School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, P.R. China
² Key Laboratory of NSLSCS, Ministry of Education, Jiangsu International Joint Laboratory of BDMCA, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R. China
³ School of Mathematical Sciences, Jiangsu Second Normal University, Nanjing 211200, P.R. China
⁴ Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA

^* Corresponding author: hfji@njupt.edu.cn; hfji1988@foxmail.com

Received: 20 December 2022
Accepted: 16 May 2023

Abstract

In this paper, an important discovery has been found for nonconforming immersed finite element (IFE) methods using the integral values on edges as degrees of freedom for solving elliptic interface problems. We show that those IFE methods without penalties are not guaranteed to converge optimally if the tangential derivative of the exact solution and the jump of the coefficient are not zero on the interface. A nontrivial counter example is also provided to support our theoretical analysis. To recover the optimal convergence rates, we develop a new nonconforming IFE method with additional terms locally on interface edges. The new method is parameter-free which removes the limitation of the conventional partially penalized IFE method. We show the IFE basis functions are unisolvent on arbitrary triangles which is not considered in the literature. Furthermore, different from multipoint Taylor expansions, we derive the optimal approximation capabilities of both the Crouzeix–Raviart and the rotated-Q₁ IFE spaces via a unified approach which can handle the case of variable coefficients easily. Finally, optimal error estimates in both H¹- and L²-norms are proved and confirmed with numerical experiments.