AIMS Mathematics, 2020, 5(5): 5240-5260. doi: 10.3934/math.2020337

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Convergence analysis and error estimate of finite element method of a nonlinear fluid-structure interaction problem

1 Department of Geographical Science and Environment Engineering, Key Laboratory of Disaster Monitoring and Mechanism Simulation of Shaanxi Province, Baoji University of Arts and Sciences, Baoji 721013, P. R. China
2 School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710129, P. R. China
3 Department of Mathematics, School of Arts and Sciences, Shaanxi University of Science and Technology, Xi’an 710021, P. R. China

In this paper, a semi-discrete finite element method for the nonlinear fluid-structure interaction problem interacts between the Navier-Stokes fluids and linear elastic solids, is studied and developed. A classical mixed variational principle of the weak formulation is given, and the corresponding finite element method is defined. As for the nonlinearity arising from the nonlinear interaction problem, we consider in time of a solution for suitably small data, and uniqueness hypothesis. This approach is fairly robust and adapts to the important case of interface with fractures or cracks. Convergence and estimate of the finite element method are also obtained for the nonlinear fluid-structure interaction problem. Finally, numerical experiments are presented to show the performance of the proposed method.
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