Guaranteed lower eigenvalue bounds for two spectral problems arising in fluid mechanics

Abstract In this paper, we obtain guaranteed lower bounds for eigenvalues of two spectral problems arising in fluid mechanics by using the min–max principles of weak form that can be derived by the principles of operator forms. These two problems are the Laplace model for fluid–solid vibrations and the sloshing problem. We deal with the positive semi-definiteness of the associated bilinear forms by adding some constraints to the solution and finite element spaces. Numerical experiments are reported finally to validate our theoretical results.