An hp finite element a posteriori analysis of a non-standard eigenvalue problem arising in fluid-solid vibrations on curved domains

In this paper we introduce and analyze an hp finite element method to solve a non-standard spectral problem in a curved plane domain using curved elements. This problem arises from nuclear engineering: the vibration of elastically mounted tubes immersed in a cavity filled with fluid. The eigenvalue problem is presented in a proper setting and we prove, under appropriate assumptions about the curved domain, the convergence of the method and a priori error estimates for the eigenfunctions and the eigenvalues. We define an efficiency and reliability a posteriori error indicator of the residual type up to higher order terms. We analyze in detail the symmetric case, and we propose and efficient approach which allows us simplify the eigenvalue problem and solve efficiently the case of multiples eigenvalues. Finally, we present an hp adaptive algorithm and some numerical tests which show the performance of the scheme, including evidence of exponential convergence.