A cheapest stable nonconforming finite element method is presented for solving the incompressible flow in a square cavity without smoothing the corner singularities. The stable cheapest nonconforming finite element pair based on P1×P0 on rectangularmeshes [29] is employed with a minimal modification of the discontinuous Dirichlet data on the top boundary, where is the finite element space of piecewise constant pressures with the globally one-dimensional checker-board pattern subspace eliminated. The proposed Stokes elements have the least number of degrees of freedom compared to those of known stable Stokes elements. Three accuracy indications for our elements are analyzed and numerically verified. Also, various numerous computational results obtained by using our proposed element show excellent accuracy.

In this paper we explore the performance of the SIMPLER, augmented Lagrangian, ‘grad-div’ preconditioners and their new variants for the two-by-two block systems arising in the incompressible Navier-Stokes equations. The lid-driven cavity and flow over a finite flat plate are chosen as the benchmark problems. For each problem the Reynolds number varies from a low to the limiting number for a laminar flow.

This paper is devoted to a new high-accuracy finite difference scheme for solving reaction-convection-diffusion problems with a small diffusivity ε. With a novel treatment for the reaction term, we first derive a difference scheme of accuracy O(εh2+εh2+h3) for the 1-D case. Using the alternating direction technique, we then extend the scheme to the 2-D case on a nine-point stencil. We apply the high-accuracy finite difference scheme to solve the 2-D steady incompressible Navier-Stokes equations in the stream function-vorticity formulation. Numerical examples are given to illustrate the effectiveness of the proposed difference scheme. Comparisons made with some high-order compact difference schemes show that the newly proposed scheme can achieve good accuracy with a better stability.

This paper presents limits for stability of projection type schemes when using high order pressure-velocity pairs of same degree. Two high order h/p variational methods encompassing continuous and discontinuous Galerkin formulations are used to explain previously observed lower limits on the time step for projection type schemes to be stable [18], when h- or p-refinement strategies are considered. In addition, the analysis included in this work shows that these stability limits do not depend only on the time step but on the product of the latter and the kinematic viscosity, which is of particular importance in the study of high Reynolds number flows. We show that high order methods prove advantageous in stabilising the simulations when small time steps and low kinematic viscosities are used.

Drawing upon this analysis, we demonstrate how the effects of this instability can be reduced in the discontinuous scheme by introducing a stabilisation term into the global system. Finally, we show that these lower limits are compatible with Courant-Friedrichs-Lewy (CFL) type restrictions, given that a sufficiently high polynomial order or a mall enough mesh spacing is selected.

In this work we introduce a new class of lowest order methods for diffusive problems on general meshes with only one unknown per element. The underlying idea is to construct an incomplete piecewise affine polynomial space with optimal approximation properties starting from values at cell centers. To do so we borrow ideas from multi-point finite volume methods, although we use them in a rather different context. The incomplete polynomial space replaces classical complete polynomial spaces in discrete formulations inspired by discontinuous Galerkin methods. Two problems are studied in this work: a heterogeneous anisotropic diffusion problem, which is used to lay the pillars of the method, and the incompressible Navier-Stokes equations, which provide a more realistic application. An exhaustive theoretical study as well as a set of numerical examples featuring different difficulties are provided.

We propose a numerical scheme to compute the motion of a two-dimensional rigid body in a viscous fluid. Our method combines the method of characteristics with a finite element approximation to solve an ALE formulation of the problem. We derive error estimates implying the convergence of the scheme.

In this paper, a Dirichlet-Neumann substructuring domaindecomposition method is presented for a finite elementapproximation to the nonlinear Navier-Stokes equations. It isshown that the Dirichlet-Neumann domain decomposition sequenceconverges geometrically to the true solution provided the Reynoldsnumber is sufficiently small. In this method, subdomain problemsare linear. Other version where the subdomain problems are linearStokes problems is also presented.