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In this paper, we apply a simple finite element numerical scheme, proposed in an earlier work (Liu in Math Comput 70(234):579-593, 2000), to perform a high resolution numerical simulation of incompressible flow over an irregular domain and analyze its boundary layer separation. Compared with many classical finite element fluid solvers, this numerical method avoids a Stokes solver, and only two Poisson-like equations need to be solved at each time step/stage. In addition, its combination with the fully explicit fourth order Runge-Kutta (RK4) time discretization enables us to compute high Reynolds number flow in a very efficient way. As an application of this robust numerical solver, the dynamical mechanism of the boundary layer separation for a triangular cavity flow with Reynolds numbers \(Re=10^4\) and \(Re=10^5\) , including the precise values of bifurcation location and critical time, are reported in this paper. In addition, we provide a super-convergence analysis for the simple finite element numerical scheme, using linear elements over a uniform triangulation with right triangles.

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In this paper, we introduce a new adaptive data analysis method to study trend and instantaneous frequency of nonlinear and nonstationary data. This method is inspired by the Empirical Mode Decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {a(t) cos(θ(t))}, where a ∈ V (θ), V (θ) consists of the functions smoother than cos(θ(t)) and θ   0. This problem can be formulated as a nonlinear l0 optimization problem. In order to solve this optimization problem, we propose a nonlinear matching pursuit method by generalizing the classical matching pursuit for the l0 optimization problem. One important advantage of this nonlinear matching pursuit method is it can be implemented very efficiently and is very stable to noise. Further, we provide an error analysis of our nonlinear matching pursuit method under certain scale separation assumptions. Extensive numerical examples will be given to demonstrate the robustness of our method and comparison will be made with the state-of-the-art methods. We also apply our method to study data without scale separation, and data with incomplete or under-sampled data.

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