The direction splitting approach proposed earlier in [7], aiming at the efficient solution of Navier-Stokes equations, is extended and adopted here to solve the Navier-Stokes-Brinkman equations describing incompressible flows in pure fluid and in porous media. The resulting pressure equation is a perturbation of the incompressibility constraint using a direction-wise factorized operator as proposed in [7]. We prove that this approach is unconditionally stable for the unsteady Navier-Stokes-Brinkman problem. We also provide numerical illustrations of the method's accuracy and effciency.

In this paper, we present an overview of the progress of the finite volume element (FVE) methods. We show that the linear FVE methods are quite mature due to their close relationship to the linear finite element methods, while development of higher order finite volume methods remains a difficult and promising research front. Theoretical analysis, as well as the algorithms and applications of these methods, are reviewed.

A mixed finite element method combined with a fixed point algorithm is proposed for solving the thermally coupled quasi-Newtonian flow problem. The existence and uniqueness of the mixed variational solution are established. A more general uniqueness result for the original system problem is presented. The convergence of the approximate solution is analyzed and the corresponding error estimates are given.

This paper presents the multiscale analysis and computation for parabolic equations with rapidly oscillating coefficients in general domains. The major contributions of this study are twofold. First, we define the boundary layer solution and the convergence rate with ε^{1/2} for the multiscale asymptotic solutions in general domains. Secondly, a highly accurate computational algorithm is developed. Numerical simulations are then carried out to validate the theoretical results.

We investigate a first order weak symplectic numerical scheme for stochastic Hamiltonian systems. Given the solution (X_t) and a class of functions f, we derive the expansion of the global approximation error for the computed E(f(X_t)) in powers of the discretization step size. The present study is an extension of the results obtained by Talay and Tubaro for the explicit Euler scheme. Based on the derived global error expansion, we construct an extrapolation method of the weak second order. The performance of the extrapolation method is demonstrated numerically for a model simulating oscillations of the particles in storage rings.