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In this paper, we study the nonconforming Crouzeix–Raviart (CR) element approximation for the fluid–solid vibration problem. We present the a priori error estimate and the a posteriori error estimate, and we prove the reliability and efficiency of the a posteriori error indicator up to higher order terms. Moreover, we carry out the numerical experiments in adaptive fashion which together with theoretical analysis show that the CR element eigenvalues approximate the exact ones from below.

The a priori and a posteriori error estimates of Crouzeix–Raviart element for the fluid–solid vibration problem