Abstract: A high order spectral element method with a domain decomposition Stokes solver is presented for hydrodynamic stability analysis.A Jacobian-free Inexact-Newton Krylov algorithm with a Stokes time-stepping preconditioning technique is introduced to the calculation of steady state of incompressible flows.An Anoldi method is used to calculate the leading eigenvalues and corresponding eigenvectors,which are responsible for the hydrodynamic instability.The method deals with steady and unsteady simulations in a similar way without time-splitting divergence error and does not need Jacobian matrix.As a result,it reduces memory allocation and computation cost,and speeds up the convergence.(Numerical) result for Kovasznay flows with an analytic solution shows spectral accuracy with exponentially spacial convergence and superlinear convergence for inexact Newton method.An antisymmetric sinusoidal velocity driven cavity problem is considered at Re=800.The stable and unstable patterns are analyzed with leading eigenvalues of steady states.The symmetric-breaking Hopf bifurcations are considered in the wake of a circular cylinder limitlessly or between two parallel walls.The onset of instabilities agrees well with experimental and numerical results.

Key words: higher order spectral element method, inexact Newton-Krylov method, Arnoldi method, hydrodynamic stability