Least-Squares Finite Element Method for the Fluid-Structure Interaction Spectral Problem

A least-squares finite element method (LSFEM) for the fluid-structure interaction problem is derived. The fluid-structure interaction problems deals with the vibration caused by an elastic structure in contact with a fluid. The fluid-structure interaction is treated as a first-order system, after which the least-squares functional is formulated using L^2-norm residuals. Norm-equivalence of the least-squares functional is proven which ensures the favorable properties of the Rayleigh-Ritz setting. As a result, a priori error estimates are obtained for the LSFEM approximation. Furthermore, we highlight steps taken towards numerical implementation and demonstrate convergence for a simplified test case. Finally, the least-squares functional is used as an inherent a posteriori error estimate for a primal displacement/pressure formulation.