A unified least-squares formulation for fluid–structure interaction problems

Abstract

In this exploratory article a new least squares formulation for the solution of fluid structure interaction problems consisting of the Navier–Stokes equations and the equations of linear elastodynamics will be analysed. It is an extension of the ideas presented in [Cai Z, Starke G. First-order system least squares for the stress–displacement formulation: linear elasticity. SIAM J Numer Anal 2003;41:715–30, Cai Z, Lee B, Wang P. Least-squares methods for incompressible newtonian fluid flow: linear stationary problem. SIAM J Numer Anal 2004;42(2):843–59] for the equations of linear elasticity and the Stokes equations. In those papers a mixed first order least squares formulation which includes the stresses as additional unknowns is proposed. As the stress field has to be approximated by discrete subspaces of H div, the usual compatibility of the normal traction will automatically be satisfied on any element edge. Therefore the strongly coupled problem can be formulated in an almost uniform manner. After introducing the basic ideas and the general formulation, a computational error analysis is presented which confirms optimal convergence rates in all problem unknowns. Then the formulation is applied to test cases which come closer to real life applications. Also for these cases the formulation achieves a convincing accuracy.