A Petrov-Galerkin immersed finite element method for steady Navier-Stokes interface problem with non-homogeneous jump conditions

This paper develops a lowest -order Petrov-Galerkin immersed Q1 - Q0 method for solving Stokes interface problem. We make use of a uniform, interface -unfitted Cartesian mesh. An immersed Petrov-Galerkin formulation is presented, where the test spaces are conventional finite element spaces and the solution spaces satisfying the jump conditions. For the Stokes and Navier-Stokes interface problem, simple stabilized items are introduced. The nonlinear convective term is treated using Picard's iteration. Extensive numerical experiments validate the feasibility and optimal convergence order for the Petrov-Galerkin immersed Q1 - Q0 scheme both with homogeneous and non -homogeneous jumps.