Convergence of successive approximation for a free-boundary problem in fluid-structure interaction

Fluid-structure-interaction problems have prominence in aerospace engineering and many other scientific and engineering disciplines. An essential property of these problems is that the interface between the fluid and the structure constitutes a free boundary. Iterative solution methods for free-boundary problems are typically based on a partitioned solution procedure: (1) the boundary-value problem(s) is (are) solved with a subset of the free-boundary conditions imposed, and (2) the free boundary is adjusted to relax the remaining free-boundary condition. This iterative procedure is referred to as successive approximation, subiteration or Picard iteration. In the present work we investigate the convergence properties of successive approximation for a model fluid-structure interaction problem, viz., the piston problem. We establish that the iteration operator is nonnormal. An important consequence of this nonnormality is that the successive approximation process can diverge before convergence occurs. The initial divergence can cause failure of the computational method despite formal stability. As such, the nonnormality induces a profound degradation in the robustness and efficiency of the subiteration method. Numerical experiments are presented to illustrate the theoretical results.