Error estimates for two‐level penalty finite volume method for the stationary Navier–Stokes equations

Two‐level penalty finite volume method for the stationary Navier–Stokes equations based on the P₁ ? P₀ element is considered in this paper. The method involves solving one small penalty Navier–Stokes problem on a coarse mesh with mesh size H = ?^{1 / 4}h^{1 / 2}, a large penalty Stokes problem on a fine mesh with mesh size h, where 0 < ? < 1 is a penalty parameter. The method we study provides an approximate solution with the convergence rate of same order as the penalty finite volume solution (u_?h,p_?h), which involves solving one large penalty Navier–Stokes problem on a fine mesh with the same mesh size h. However, our method can save a large amount of computational time. Copyright © 2013 John Wiley & Sons, Ltd.     

Bivariate spline method for numerical solution of time evolution Navier‐Stokes equations over polygons in stream function formulation

We use a bivariate spline method to solve the time evolution Navier‐Stokes equations numerically. The bivariate splines we use in this article are in the spline space of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the Navier‐Stokes equations is employed. Galerkin's method is applied to discretize the space variables of the nonlinear fourth‐order equation, Crank‐Nicholson's method is applied to discretize the time variable, and Newton's iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in L₂(0, T; H²(Ω)) ∩ L_∞(0, T; H¹(Ω)) of the 2D nonlinear fourth‐order problem and give an estimate of how fast the numerical solution converges to the weak solution. The C¹ cubic splines are implemented in MATLAB for solving the Navier‐Stokes equations numerically. Our numerical experiments show that the method is effective and efficient. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 776–827, 2003.     

Finite volume method based on stabilized finite elements for the nonstationary Navier–Stokes problem

A finite volume method based on stabilized finite element for the two‐dimensional nonstationary Navier–Stokes equations is investigated in this work. As in stabilized finite element method, macroelement condition is introduced for constructing the local stabilized formulation of the nonstationary Navier–Stokes equations. Moreover, for P₁ ? P₀ element, the H¹ error estimate of optimal order for finite volume solution (u_h,p_h) is analyzed. And, a uniform H¹ error estimate of optimal order for finite volume solution (u_h, p_h) is also obtained if the uniqueness condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods

In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, Q ₁^rot and EQ ₁^rot. Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. This project is supported in part by the National Natural Science Foundation of China (10471103) and is subsidized by the National Basic Research Program of China under the grant 2005CB321701.     

A multilevel finite element method in space‐time for the Navier‐Stokes problem

A multilevel finite element method in space‐time for the two‐dimensional nonstationary Navier‐Stokes problem is considered. The method is a multi‐scale method in which the fully nonlinear Navier‐Stokes problem is only solved on a single coarsest space‐time mesh; subsequent approximations are generated on a succession of refined space‐time meshes by solving a linearized Navier‐Stokes problem about the solution on the previous level. The a priori estimates and error analysis are also presented for the J‐level finite element method. We demonstrate theoretically that for an appropriate choice of space and time mesh widths: h_j ～ h, k_j ～ k, j = 2, …, J, the J‐level finite element method in space‐time provides the same accuracy as the one‐level method in space‐time in which the fully nonlinear Navier‐Stokes problem is solved on a final finest space‐time mesh. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions

In this paper, we consider low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions. The time discretization is based on the Euler implicit scheme, and the spatial discretization is based on the low‐order element (P₁−P₁ or P₁−P₀) for the approximation of the velocity and pressure. Moreover, some error estimates for the numerical solution of fully discrete stabilized finite element scheme are obtained. Finally, numerical experiments are performed to confirm our theoretical results.     

Uncoupling evolutionary groundwater‐surface water flows using the Crank–Nicolson Leapfrog method

Consider an incompressible fluid in a region Ω_f flowing both ways across an interface into a porous media domain Ω_p saturated with the same fluid. The physical processes in each domain have been well studied and are described by the Stokes equations in the fluid region and the Darcy equations in the porous media region. Taking the interfacial conditions into account produces a system with an exactly skew symmetric coupling. Spatial discretization by finite element method and time discretization by Crank–Nicolson LeapFrog give a second‐order partitioned method requiring only one Stokes and one Darcy subphysics and subdomain solver per time step for the fully evolutionary Stokes‐Darcy problem. Analysis of this method leads to a time step condition sufficient for stability and convergence. Numerical tests verify predicted rates of convergence; however, stability tests reveal the problem of growth of numerical noise in unstable modes in some cases. In such instances, the addition of time filters adds stability. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Multi-level spectral Galerkin method for the Navier–Stokes equations, II: time discretization

A fully discrete multi-level spectral Galerkin method in space–time for the two-dimensional nonstationary Navier–Stokes problem is considered. The method is a multi-scale method in which the fully nonlinear Navier–Stokes problem is only solved on the lowest-dimensional space with the largest time step Δt ₁; subsequent approximations are generated on a succession of higher-dimensional spaces with small time step Δt _j by solving a linearized Navier–Stokes problem about the solution on the previous level. Some error estimates are also presented for the J-level spectral Galerkin method. The scaling relations of the dimensional numbers and time mesh widths that lead to optimal accuracy of the approximate solution in H ¹-norm and L ²-norm are investigated, i.e., m _j∼m _j−1 ^3/2 , Δt _j∼Δt _j−1 ^3/2 , j=2,. . .,J. We demonstrate theoretically that a fully discrete J-level spectral Galerkin method is significantly more efficient than the standard one-level spectral Galerkin method. Mathematics subject classifications (2000) 35L70, 65N30, 76D06 Subsidized by the Special Funds for Major State Basic Research Projects G1999032801-07, NSF of China 10371095 and the City University of Hong Kong Research Project 7001093, NSF of China 50323001.     

A finite element method for the axisymmetric three‐field Stokes system with a discontinuous pressure

A quadrilateral based velocity‐pressure‐extrastress tensor mixed finite element method for solving the three‐field Stokes system in the axisymmetric case is studied. The method derived from Fortin's Q₂ − P₁ velocity‐pressure element is to be used in connection with the standard Galerkin formulation. This makes it particularly suitable for the numerical simulation of viscoelastic flow. It is proven to be second‐order convergent in the natural weighted Sobolev norms, for the system under consideration. The crucial result that the method is uniformly stable is proven for the case of rectangular meshes. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 739–763, 1999     

Analysis of newton multilevel stabilized finite volume method for the three‐dimensional stationary Navier‐Stokes equations

This article proposes and analyzes a multilevel stabilized finite volume method(FVM) for the three‐dimensional stationary Navier–Stokes equations approximated by the lowest equal‐order finite element pairs. The method combines the new stabilized FVM with the multilevel discretization under the assumption of the uniqueness condition. The multilevel stabilized FVM consists of solving the nonlinear problem on the coarsest mesh and then performs one Newton correction step on each subsequent mesh thus only solving one large linear systems. The error analysis shows that the multilevel‐stabilized FVM provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the stationary Navier–Stokes equations on a fine mesh for an appropriate choice of mesh widths: h_j ～ h_j‐1², j = 1,…,J. Therefore, the multilevel stabilized FVM is more efficient than the standard one‐level‐stabilized FVM. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A note on a lowest order divergence‐free Stokes element on quadrilaterals

This short note reports a lowest order divergence‐free Stokes element on quadrilateral meshes. The velocity space is based on a P₁ spline element over the crisscross partition of a quadrilateral, and the pressure is approximated by piecewise constant. For a given quadrilateral mesh, this element is stable if and only if the well‐known Q₁‐P₀ element is also stable. Although this method is a subspace method of Qin‐Zhang's P₁‐P₀ element, their velocity solutions are precisely equal. Moreover, an explicit basis representation is also provided. These theoretical findings are verified by numerical tests.     

Stokes型积分-微分方程Q_2-P_1元的超收敛分析

本文研究Q2-P1混合元对Stokes型积分-微分方程的有限元方法.利用积分恒等式技巧给出了关于流体速度u和压力p的误差估计,特别是在压力p的误差中去掉了影响解的稳定性的1因子t-2,改善了以往文献的结果.同时,通过构造适当的插值后处理算子得到了整体超收敛结果.     

Time delay and finite differences for the non-stationary non-linear Navier–Stokes equations

In the present paper we use a time delay ? > 0 for an energy conserving approximation of the non-linear term of the non-stationary Navier–Stokes equations. We prove that the corresponding initial-value problem (N_?) in smoothly bounded domains G ? ?³ is well-posed. We study a semidiscretized difference scheme for (N_?) and prove convergence to optimal order in the Sobolev space H²(G). Passing to the limit ?→0 we show that the sequence of stabilized solutions has an accumulation point such that it solves the Navier–Stokes problem (N_o) in a weak sense (Hopf).     

An algebraic multigrid method for Q2−Q1 mixed discretizations of the Navier–Stokes equations

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily colocated at mesh points. Specifically, we investigate a Q ₂? Q ₁ mixed finite element discretization of the incompressible Navier–Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees of freedom (DOFs) are defined at spatial locations where there are no corresponding pressure DOFs. Thus, AMG approaches leveraging this colocated structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocity DOF relationships of the Q ₂? Q ₁ discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity DOFs resembles that on the finest grid. To define coefficients within the intergrid transfers, an energy minimization AMG (EMIN‐AMG) is utilized. EMIN‐AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier–Stokes problems.     

Stokes slip flow between corrugated walls

Stokes flow between corrugated plates in microdomains has been analyzed using a perturbation method. This approach used the incompressible Navier-Stokes equations, but the velocity-slip is present along the solid-fluid interface. For the slip flow regime, if we introduce Knudsen number (K _n) herein, 0.01 K _n 0.1, the total flow rate is increasing as a ratio of 1 + 6K _nto no-slip Stokes flow. If we consider fixedK _ncases, the corrugations still decrease the flow rate, consideringO(²) terms, and the decrease is maximum as the phase shift becomes 180 °.     

A new stabilized finite volume method for the stationary Stokes equations

In this paper we develop and study a new stabilized finite volume method for the two-dimensional Stokes equations. This method is based on a local Gauss integration technique and the conforming elements of the lowest-equal order pair (i.e., the P ₁–P ₁ pair). After a relationship between this method and a stabilized finite element method is established, an error estimate of optimal order in the H ¹-norm for velocity and an estimate in the L ²-norm for pressure are obtained. An optimal error estimate in the L ²-norm for the velocity is derived under an additional assumption on the body force. This work is supported in part by the NSF of China 10701001 and by the US National Science Foundation grant DMS-0609995 and CMG Chair Funds in Reservoir Simulation.     

The stability of rarefaction wave for Navier–Stokes equations in the half‐line

This paper studies the stability of the rarefaction wave for Navier–Stokes equations in the half‐line without any smallness condition. When the boundary value is given for velocity u∥_{x = 0} = u_? and the initial data have the state (v₊, u₊) at x→ + ∞, if u_?<u₊, it is excepted that there exists a solution of Navier–Stokes equations in the half‐line, which behaves as a 2‐rarefaction wave as t→ + ∞. Matsumura–Nishihara have proved it for barotropic viscous flow (Quart. Appl. Math. 2000; 58:69–83). Here, we generalize it to the isentropic flow with more general pressure. Copyright © 2011 John Wiley & Sons, Ltd.     

Penalty finite element approximations for the Stokes equations by L2 projection

This paper considers the penalty finite element method for the Stokes equations, based on some stable finite elements space pair (X_h, M_h) that do satisfy the discrete inf–sup condition. Theoretical results show that the penalty error converges as fast as one should expect from the order of the elements. Moreover, the penalty finite element method by L² projection can improve the penalty error estimates. Finally, we confirm these results by a series of numerical experiments. Copyright © 2008 John Wiley & Sons, Ltd.     

A lowest order divergence-free finite element on rectangular grids

It is shown that the conforming Q _2,1;1,2-Q′₁ mixed element is stable, and provides optimal order of approximation for the Stokes equations on rectangular grids. Here, Q _2,1;1,2 = Q _2,1 × Q _1,2, and Q _2,1 denotes the space of continuous piecewise-polynomials of degree 2 or less in the x direction but of degree 1 in the y direction. Q′₁ is the space of discontinuous bilinear polynomials, with spurious modes filtered. To be precise, Q′₁ is the divergence of the discrete velocity space Q _2,1;1,2. Therefore, the resulting finite element solution for the velocity is divergence-free pointwise, when solving the Stokes equations. This element is the lowest order one in a family of divergence-free element, similar to the families of the Bernardi-Raugel element and the Raviart-Thomas element.     

Two-level stabilized nonconforming finite element method for the Stokes equations

In this article, we present a new two-level stabilized nonconforming finite elements method for the two dimensional Stokes problem. This method is based on a local Gauss integration technique and the mixed nonconforming finite element of the NCP ₁ — P ₁ pair (nonconforming linear element for the velocity, conforming linear element for the pressure). The two-level stabilized finite element method involves solving a small stabilized Stokes problem on a coarse mesh with mesh size H and a large stabilized Stokes problem on a fine mesh size h = H/3. Numerical results are presented to show the convergence performance of this combined algorithm.