In two and three dimensions, this study is focused on the numerical analysis of an eigenproblem associated with a fluid-structure model for sloshing and elasto-acoustic vibration. We use a displacement-Herrmann pressure formulation for the solid, while for the fluid, a pure displacement formulation is considered. Under this approach we propose a non conforming locking-free method based on classic finite elements to approximate the natural frequencies (of the eigenmodes) of the coupled system. Employing the theory for non-compact operators we prove convergence and error estimates. Also we propose an a posteriori error estimator for this coupled problem which is shown to be efficient and reliable. All the presented theory is contrasted with a set of numerical tests in 2D and 3D.

翻译：本研究在二维和三维空间中，针对与流体-结构相互作用模型相关的特征值问题进行数值分析，该模型适用于晃动及弹性声学振动问题。固体部分采用位移-Herrmann压力公式，流体部分则采用纯位移公式。基于此框架，我们提出一种基于经典有限元的非协调无闭锁方法，用于近似耦合系统的特征频率（对应特征模态）。运用非紧算子理论，我们证明了方法的收敛性并给出了误差估计。同时，针对该耦合问题提出了一个后验误差估计子，并验证了其有效性与可靠性。所有理论结果均通过二维与三维数值实验进行了验证。