Inf-sup stability of Petrov-Galerkin immersed finite element methods for one-dimensional elliptic interface problems

In this article, we analyze the Petrov-Galerkin immersed finite element method (PG-IFEM) when applied to one-dimensional elliptic interface problems. In the PG-IFEM (T. Hou, X. Wu and Y. Zhang, Commun. Math. Sci., 2 (2004), 185-205, and S. Hou and X. Liu, J. Comput. Phys., 202 (2005), 411-445), the classic immersed finite element (IFE) space was taken as the trial space while the conforming linear finite element space was taken as the test space. We first prove the inf-sup condition of the PG-IFEM and then show the optimal error estimate in the energy norm. We also show the optimal estimate of the condition number of the stiffness matrix. The results are extended to two dimensional problems in a special case.