Error-amplification analysis of subiteration-preconditioned GMRES for fluid–structure interaction

The customary subiteration method for solving fluid–structure-interaction problems exhibits several deficiencies, viz., only conditional stability, potential convergence difficulties due to non-normality-induced divergence, and the inability to reuse information from previously solved similar problems. To overcome these deficiencies, a novel solution method is considered, in which subiteration is used as a preconditioner to GMRES. This paper treats the linear-algebra aspects of the subiteration method, and of the subiteration-preconditioned GMRES method, on the basis of properties of the error-amplification matrix for the aggregated fluid–structure system. An analysis of the error-amplification matrix of subiteration establishes that subiteration condenses errors into a low-dimensional subspace which can be associated with the interface degrees-of-freedom. Therefore, the GMRES acceleration of subiteration can be confined to the interface degrees-of-freedom. The error-amplification analysis provides a clear explanation of the relation between the local GMRES acceleration (i.e., on the interface degrees-of-freedom), and the global error-amplification properties (i.e., for the aggregated system). Moreover, we show that the subiteration iterates span a Krylov space corresponding to a preconditioned aggregated system. We then address the implications of the non-normality of the subiteration preconditioner for the convergence of GMRES. The subiteration-preconditioned GMRES method enables the optional reuse of Krylov vectors in subsequent invocations of GMRES, which can substantially enhance the efficiency of the method. To assess the potential and the limitations of the reuse option, we analyse the error-amplification matrix of the GMRES method with reuse. Furthermore, we establish that the GMRES acceleration on the interface degrees-of-freedom generates an approximation to the Schur complement for the aggregated system. The GMRES acceleration and the reuse of Krylov vectors are then assessed in terms of the approximation properties for the Schur complement, and in terms of the properties of the corresponding error-amplification matrices. Numerical experiments on a model fluid–structure-interaction problem illustrate the developed theory. In particular, we analyse the convergence of the respective methods in terms of spectral radii, matrix norms and sharp convergence upper bounds.