A new parameter free partially penalized immersed finite element and the optimal convergence analysis

This paper presents a new parameter free partially penalized immersed finite element method and convergence analysis for solving second order elliptic interface problems. The optimal approximation capabilities of the immersed finite element space is proved via a novel new approach that is much simpler than that in the literature. A new trace inequality which is necessary to prove the optimal convergence of immersed finite element methods is established on interface elements. Optimal error estimates are derived rigorously even though the curved interface is approximated by line segments. The new method and analysis have also been extended to problems with variable coefficients. Numerical examples are also provided to confirm the theoretical analysis and efficiency of the new method.