Fully discrete finite element method based on pressure stabilization for the transient Stokes equations

In this work, a new fully discrete stabilized finite element method is studied for the two-dimensional transient Stokes equations. This method is to use the difference between a consistent mass matrix and underintegrated mass matrix as the complement for the pressure. The spatial discretization is based on the P₁–P₁ triangular element for the approximation of the velocity and pressure, the time discretization is based on the Euler semi-implicit scheme. Some error estimates for the numerical solutions of fully discrete stabilized finite element method are derived. Finally, we provide some numerical experiments, compared with other methods, we can see that this novel stabilized method has better stability and accuracy results for the unsteady Stokes problem.     

Multiscale enrichment of a finite volume element method for the stationary Navier–Stokes problem

In this paper, we introduce a finite volume element method for the Navier–Stokes problem. This method is based on the multiscale enrichment and uses the lowest finite element pair P ₁/P ₀. The stability and convergence of the optimal order in H ¹-norm for velocity and L ²-norm for pressure are obtained. Using a dual problem for the Navier–Stokes problem, we establish the convergence of the optimal order in L ²-norm for the velocity.     

Two-level Stabilized Finite Element Methods for the Steady Navier–Stokes Problem

In this article, the two-level stabilized finite element formulations of the two-dimensional steady Navier–Stokes problem are analyzed. A macroelement condition is introduced for constructing the local stabilized formulation of the steady Navier–Stokes problem. By satisfying this condition the stability of the Q₁–P₀ quadrilateral element and the P₁–P₀ triangular element are established. Moreover, the two-level stabilized finite element methods involve solving one small Navier–Stokes problem on a coarse mesh with mesh size H, a large Stokes problem for the simple two-level stabilized finite element method on a fine mesh with mesh size h=O(H²) or a large general Stokes problem for the Newton two-level stabilized finite element method on a fine mesh with mesh size h=O(|log h|^1/2H³). The methods we study provide an approximate solution (u^h,p^h) with the convergence rate of same order as the usual stabilized finite element solution, which involves solving one large Navier–Stokes problem on a fine mesh with mesh size h. Hence, our methods can save a large amount of computational time.     

Pressure projection stabilized finite element method for Navier–Stokes equations with nonlinear slip boundary conditions

In this paper, we consider the pressure projection stabilized finite element method for the Navier–Stokes equation with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with Navier–Stokes operator. The H ¹ and L ² error estimates for the velocity and the L ² error estimate for the pressure are obtained. Finally, the numerical results are displayed to verify the theoretical analysis.     

A Stabilized Nitsche Fictitious Domain Method for the Stokes Problem

We present a novel finite element method for the Stokes problem on fictitious domains. We prove inf-sup stability, optimal order convergence and uniform boundedness of the condition number of the discrete system. The finite element formulation is based on a stabilized Nitsche method with ghost penalties for the velocity and pressure to obtain stability in the presence of small cut elements. We demonstrate for the first time the applicability of the Nitsche fictitious domain method to three-dimensional Stokes problems. We further discuss a general, flexible and freely available implementation of the method and present numerical examples supporting the theoretical results.     

Non-nested multi-grid solvers for mixed divergence-free Scott–Vogelius discretizations

We apply the general framework developed by John et al. in Computing 64:307–321, 2000 to analyze the convergence of multi-level methods for mixed finite element discretizations of the generalized Stokes problem using the Scott–Vogelius element. The Scott–Vogelius element seems to be promising since discretely divergence-free functions are divergence-free pointwise. However, to satisfy the Ladyzhenskaya–Babu?ka–Brezzi stability condition, we have to deal in the multi-grid analysis with non-nested families of meshes which are derived from nested macro element triangulations. Additionally, the analysis takes into account an optional symmetric stabilization operator which suppresses spurious oscillations of the velocity provoked by a dominant reaction term. Usually, the generalized Stokes problems appears in semi-implicit splitting schemes for the unsteady Navier–Stokes equations, but the symmetric part of a stabilized discrete Oseen problem can be reguarded as a discrete generalized Stokes problem likewise.     

Two-level stabilized finite element method for the transient Navier–Stokes equations

In this article, a two-level stabilized finite element method based on two local Gauss integrations for the two-dimensional transient Navier–Stokes equations is analysed. This new stabilized method presents attractive features such as being parameter-free, or being defined for nonedge-based data structures. Some new a priori bounds for the stabilized finite element solution are derived. The two-level stabilized method involves solving one small Navier–Stokes problem on a coarse mesh with mesh size 0<H<1, and a large linear Stokes problem on a fine mesh with mesh size 0<h?H. A H ¹-optimal velocity approximation and a L ²-optimal pressure approximation are obtained. If we choose h=O(H ²), the two-level method gives the same order of approximation as the standard stabilized finite element method.     

Equal-order finite elements with local projection stabilization for the Darcy–Brinkman equations

For the Darcy–Brinkman equations, which model porous media flow, we present an equal-order H¹-conforming finite element method for approximating velocity and pressure based on a local projection stabilization technique. The method is stable and accurate uniformly with respect to the coefficients of the viscosity and the zeroth order term in the momentum equation. We prove a priori error estimates in a mesh-dependent norm as well as in the L²-norm for velocity and pressure. In particular, we obtain optimal order of convergence in L² for the pressure in the Darcy case with vanishing viscosity and for the velocity in the general case with a positive viscosity coefficient. Numerical results for different values of the coefficients in the Darcy–Brinkman model are presented which confirm the theoretical results and indicate nearly optimal order also in cases which are not covered by the theory.     

A priori error estimation of a four-node Reissner–Mindlin plate element for elasto-dynamics

The explicit finite element method for transient dynamics of linear elasticity by Reissner–Mindlin plate model is introduced. For clamped rectangular plate, the a priori error estimates are derived for the four-node Bathe–Dvorkin element. For fixed thickness, the convergence rates of deflection, rotation, and their velocities, measured both in H¹-norm and L²-norm, can possibly all be optimal under certain conditions. In some cases, the numerical examples show that the convergence rate stays optimal for a certain range of thickness. In other cases, however, the deterioration in rate of convergence and even locking may occur to the velocity terms.     

A stabilized finite volume method for the evolutionary Stokes–Darcy system

In this paper, we propose a stabilized fully discrete finite volume method based on two local Gauss integrals for a non-stationary Stokes–Darcy problem. This stabilized method is free of stabilized parameters and uses the lowest equal-order finite element triples $P1\text{–}P1\text{–}P1$ for approximating the velocity, pressure and hydraulic head of the Stokes–Darcy model. Under a modest time step restriction in relation to physical parameters, we give the stability analysis and the error estimates for the stabilized finite volume scheme by means of a relationship between finite volume and finite element approximations with the lower order elements. Finally, a series of numerical experiments are provided to demonstrate the validity of the theoretical results.     

A numerical computation of a 3-D stokes problem in the description of intestinal peristaltic waves

The mathematical aspects of a physical model of intestinal peristaltic waves involve numerical methods for calculating and simulating the solution. A finite element method is applied to Stokes' equations in R³ in order to calculate the velocity-pressure couple at each vertex of the pentahedron elements of the geometrical model domain, for viscous and non-compressible fluid. By using special computational programs, the domain may be stitched to create a grid domain in R³. The finite reference element of the grid is a pentahedron. The resultant linear system may be solved by applying Gauss' direct method to obtain velocities and pressures at each grid point. Graphical representations of velocity profiles in R³, show positive and negative zones for the output, with positions that vary from one geometrical model to another. The calculated velocity and pressure values are shown to change according to the applied vector efforts. The numerical results obtained are in good agreement with experimental data. This suggests that the finite element method is useful for solving Stokes' equations to describe intestinal peristalsis waves.     

Covolume method for new nonconforming rectangle element for the Stokes problem

We analyze a covolume method based on the new nonconforming element introduced by Douglas et al. [1]. We show the H¹ optimal order convergence of the scheme for Stokes problem and study the hybrid domain decomposition procedure for this covolume scheme. The numerical experiment shows that the covolume scheme is somewhat better than finite element scheme in the computation of pressure.     

Numerical analysis and computations of a high accuracy time relaxation fluid flow model

We present a numerical study based on continuous finite element analysis for a time relaxation regularization of Navier–Stokes equations. This regularization is based on filtering and deconvolution. We study the convergence of the regularized equations using a fully discretized filter and deconvolution algorithm. Velocity and pressure error estimates and the L ₂ Aubin–Nitsche lift technique are proved for the equilibrium problem, and this analysis is accompanied by the velocity error estimate for the time-dependent problem, too. Thus, optimal error estimates in L ₂ and H ¹ norms are derived and followed by their computational verification. Also, computational results of the vortex street are presented for the two-dimensional cylinder benchmark flow problem. Maximum drag and lift coefficients and difference in pressure between the front and back of the cylinder at the final time were investigated as well, showing that the time relaxation regularization can attain the benchmark values.     

Direct numerical simulation of turbulent channel flows using a stabilized finite element method

Direct numerical simulations (DNS) of incompressible turbulent channel flows at Re_τ = 180 and 395 (i.e., Reynolds number, based on the friction velocity and channel half-width) were performed using a stabilized finite element method (FEM). These simulations have been motivated by the fact that the use of stabilized finite element methods for DNS and LES is fairly recent and thus the question of how accurately these methods capture the wide range of scales in a turbulent flow remains open. To help address this question, we present converged results of turbulent channel flows under statistical equilibrium in terms of mean velocity, mean shear stresses, root mean square velocity fluctuations, autocorrelation coefficients, one-dimensional energy spectra and balances of the transport equation for turbulent kinetic energy. These results are consistent with previously published DNS results based on a pseudo-spectral method, thereby demonstrating the accuracy of the stabilized FEM for turbulence simulations.     

Pressure bubbles stabilization features in the Stokes problem

The standard finite element approximation using equal-order-linear-continuous velocity–pressure variables is enriched with velocity and pressure bubble functions to model the Stokes problem. We show by static condensation that these bubble functions give rise to a stabilized method involving least-squares forms of the momentum and of the continuity equations. In particular, pressure bubbles play a key role in explaining the addition of the least-squares form of the continuity equation in a stabilized method for Stokes.     

Parallel two-grid finite element method for the time-dependent natural convection problem with non-smooth initial data

In this paper, we consider the parallel two-grid finite element method for the transient natural convection problem with non-smooth initial data. Our numerical scheme involves solving a nonlinear natural convection problem on the coarse grid and solving a linear natural convection problem on the fine grid. The linear natural convection problem can be split into two subproblems which can be solved in parallel: a linearized Navier–Stokes problem and a linear parabolic problem. We firstly provide the stability and convergence of standard Galerkin finite element method with non-smooth initial data. Secondly, we develop optimal error estimates of two-grid finite element method for velocity and temperature in $H^1$-norm and for pressure in $L^2$-norm. In order to overcome the difficulty posed by the loss of regularity, some suitable weight functions are introduced in our stability and convergence analysis for the natural convection equations. Finally, some numerical results are presented to verify the established theoretical results.     

Weighted least-squares finite element methods for the linearized Navier–Stokes equations

We implemented weighted least-squares finite element methods for the linearized Navier-Stokes equations based on the velocity–pressure–stress and the velocity–vorticity–pressure formulations. The least-squares functionals involve the L²-norms of the residuals of each equation multiplied by the appropriate weighting functions. The weights included a mass conservation constant, a mesh-dependent weight, a nonlinear weighting function, and Reynolds numbers. A feature of this approach is that the linearized system creates a symmetric and positive-definite linear algebra problem at each Newton iteration. We can prove that least-squares approximations converge with the linearized version solutions of the Navier–Stokes equations at the optimal convergence rate. Model problems considered in this study were the flow past a planar channel and 4-to-1 contraction problems. We presented approximate solutions of the Navier–Stokes problems by solving a sequence of the linearized Navier–Stokes problems arising from Newton iterations, revealing the convergence rates of the proposed schemes, and investigated Reynolds number effects.     

A priori and a posteriori error analysis for a large-scale ocean circulation finite element model

We consider the finite element solution of the stream function–vorticity formulation for a large-scale ocean circulation model. First, we study existence and uniqueness of solution for the continuous and discrete problems. Under appropriate regularity assumptions we prove that the stream function can be computed with an error of order h in H¹-seminorm. Second, we introduce and analyze an h-adaptive mesh refinement strategy to reduce the spurious oscillations and poor resolution which arise when convective terms are dominant. We propose an a posteriori anisotropic error indicator based on the recovery of the Hessian from the finite element solution, which allows us to obtain well adapted meshes. The numerical experiments show an optimal order of convergence of the adaptive scheme. Furthermore, this strategy is efficient to eliminate the oscillations around the boundary layer.     

Superconvergence of a Mixed Covolume Method for Elliptic Problems

We consider a mixed covolume method for a system of first order partial differential equations resulting from the mixed formulation of a general self-adjoint elliptic problem with a variable full diffusion tensor. The system can be used to model the transport of a contaminant carried by a flow. We use the lowest order Raviart-Thomas mixed finite element space. We show the first order convergence in L ² norm and the superconvergence in certain discrete norms both for the pressure and velocity. Finally some numerical examples illustrating the error behavior of the scheme are provided. Supported by the National Natural Science Foundation of China under grant No. 10071044 and the Research Fund of Doctoral Program of High Education by State Education Ministry of China.