Fully discrete error estimation for a quasi-Newtonian fluid-structure interaction problem

Fluid-structure interaction (FSI) problems have various applications in engineering and biology, where two dynamics, fluid flow and deformable structure, are considered in one system. Simulations of blood flow, tidal current turbines and gas explosions in pipelines are some well-known applications in this research area [2, 6, 9, 16, 20, 21]. FSI problems concern the mutual influence between two dynamics: the domain of fluid is determined by structure deformation, and structure movement is determined by fluid stress. The numerical discretization of an FSI system poses great computational challenges due to the nature of its complexity. A fully-coupled scheme, which solves the fluid and structure subproblems simultaneously, results in a large system, which in turn requires large memory storage and a special solver. However, the monolithic approach has been used widely, in particular for blood flow problems, where a stability issue caused by the added-mass effect exists in many partitioned algorithms [5, 11, 14, 15, 22]. Recently, we used a monolithic approach to investigate the finite element approximation of a quasi-Newtonian FSI problem [18]. The Arbitrary Lagrangian Eulerian method (ALE) was used to deal with the timedependent domain of the fluid. In the ALE method, an invertible and sufficiently regular ALE mapping is introduced to obtain a conforming mesh at arbitrary time following the interface movement as the image of a fixed mesh in the reference domain. We analyzed a semi-discrete FSI system written in the ALE frame for stability, proved an error estimate and performed numerical tests. Results have shown the standard optimal convergence rate