## Preprint 2012-09

## Claudio Padra, Rodolfo Rodríguez, Mario Scheble:

### An hp finite element method to solve a fluid-solid vibration problem

### Abstract:

This paper deals with a two-dimensional fluid-solid vibration problem arising from nuclear engineering: the vibration of elastically mounted tubes immersed in a cavity filled with fluid. A convenient variational formulation of this problem, valid for compressible and incompressible fluids, is introduced. An hp finite element method is used for its discretization, which leads to a well posed matrix eigenvalue problem. Optimal order a priori error estimates are proved for eigenfunctions and eigenvalues. Then, local a posteriori error indicators are defined and its efficiency and reliability are studied. An adaptive scheme driven by these indicators is proposed and numerically tested.

This preprint gave rise to the following definitive publication(s):

**Claudio PADRA, Rodolfo RODRíGUEZ, Mario SCHEBLE: ***An hp finite element method to solve a fluid-solid vibration problem*. SIAM Journal on Scientific Computing, vol. 34, 5, pp. A2533-A2557, (2012).