Semi and fully discrete error analysis for elastodynamic interface problems using immersed finite element methods. (English) Zbl 1538.65348
Summary: 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 \(\mathcal{P}_1\) and \(\mathcal{Q}_1\) 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, \(L^2\) and semi-\(H^1\) norms. Numerical examples are designed to verify our theoretical analysis and to demonstrate the stability and robustness of our schemes.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 74A50 | Structured surfaces and interfaces, coexistent phases |
| 74H15 | Numerical approximation of solutions of dynamical problems in solid mechanics |
| 74B10 | Linear elasticity with initial stresses |
| 74S20 | Finite difference methods applied to problems in solid mechanics |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
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