​An Iterative Approach for Constructing Immersed Finite Element Spaces and Applications to Interface Problems

Abstract

In this paper, an iterative approach for constructing immersed finite element spaces is developed for various interface conditions of interface problems involving multiple primary variables. Combining such iteratively constructed immersed finite element spaces with the distributed Lagrange multiplier/fictitious domain (DLM/FD) method, we further present a new discretization method that can uniformly solve general interface problems with multiple primary variables and/or with different governing equations on either side of the interface, including fluid-structure interaction problems. The strengths of the proposed method are shown in the numerical experiments for Stokes- and Stokes/elliptic interface problems with different types of interface conditions, where, the optimal or nearly optimal convergence rates are obtained for the velocity variable in $H^1$, $L^2$ and $L$^∞ norms, and at least 1.5-th order convergence for the pressure variable in $L^2$ norm within few number of iterations. In addition, numerical experiments show that such iterative process uniformly converges and the number of iteration is independent of mesh ratios and jump ratios.