Semi and fully discrete error analysis for elastodynamic interface problems using immersed finite element methods

In this paper, we present an immersed finite element (IFE) method for solving the elastodynamics interface problems on interface-unfitted meshes. For spatial discretization, we use vector-valued P1 and Q1 IFE spaces. We establish some important properties of these IFE spaces, such as inverse inequalities, which will be crucial in the error analysis. For temporal discretization, both the semi-discrete and the fully discrete schemes are derived. The proposed schemes are proved to be unconditionally stable and enjoy optimal rates of convergence in the energy, L2 and semi-H1 norms. Numerical examples are designed to verify our theoretical analysis and to demonstrate the stability and robustness of our schemes.