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.

In this paper we discuss the derivation and use of local pressure boundary conditions for finite difference schemes for the unsteady incompressible Navier-Stokes equations in the velocity-pressure formulation. Their use is especially well suited for the computation of moderate to large Reynolds number flows. We explore the similarities between the implementation and use of local pressure boundary conditions and local vorticity boundary conditions in the design of numerical schemes for incompressible flow in 2D. In their respective formulations, when these local numerical boundary conditions are coupled with a fully explicit convectively stable time stepping procedure, the resulting methods are simple to implement and highly efficient. Unlike the vorticity formulation, the use of the local pressure boundary condition approach is readily applicable to 3D flows. The simplicity of the local pressure boundary condition approach and its easy adaptation to more general flow settings make the resulting scheme an attractive alternative to the more popular methods for solving the Navier-Stokes equations in the velocity-pressure formulation. We present numerical results of a second-order finite difference scheme on a nonstaggered grid using local pressure boundary conditions. Stability and accuracy of the scheme applied to Stokes flow is demonstrated using normal mode analysis. Also described is the extension of the method to variable density flows.

A very simple and efficient finite element method is introduced for two and three dimensional viscous incompressible flows using the vorticity formulation. This method relies on recasting the traditional finite element method in the spirit of the high order accurate finite difference methods introduced by the authors in another work. Optimal accuracy of arbitrary order can be achieved using standard finite element or spectral elements. The method is convectively stable and is particularly suited for moderate to high Reynolds number flows.

The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be viewed, in particular, as solution sets to problems of generalized semi-infinite programming with polynomial data. Exploiting the imposed polynomial structure together with powerful tools of variational analysis and semialgebraic geometry, we establish a far-going extension of the Łojasiewicz gradient inequality to the general nonsmooth class of supremum marginal functions as well as higher-order (Hölder type) local error bounds results with explicitly calculated exponents. The obtained results are applied to higher-order quantitative stability analysis for various classes of optimization problems including generalized semi-infinite programming with polynomial data, optimization of real polynomials under polynomial matrix inequality constraints, and polynomial second-order cone programming. Other applications provide explicit convergence rate estimates for the cyclic projection algorithm to find common points of convex sets described by matrix polynomial inequalities and for the asymptotic convergence of trajectories of subgradient dynamical systems in semialgebraic settings.

We use a small positive parameter to change the totalvariation function for unconstrainedMRimage reconstruction to a strictly convex perturbed function. Bregman iteration is applied to solve the modified total-variation MR image (TVMRI) reconstruction problem. A lagged diffusivity fixed-point algorithm is applied to solve the minimization problem in the Bregman iteration. We use the periodic boundary condition and a Fourier transform to accelerate TVMRI reconstruction.RealMR images are used to test the approach in numerical experiments. The experimental results demonstrate that the proposed method is very efficient for TVMRI reconstruction.