A stabilized finite element method for convection–diffusion problems

A stabilized finite element method (FEM) is presented for solving the convection–diffusion equation. We enrich the linear finite element space with local functions chosen according to the guidelines of the residual‐free bubble (RFB) FEM. In our approach, the bubble part of the solution (the microscales) is approximated via an adequate choice of discontinuous bubbles allowing static condensation. This leads to a streamline‐diffusion FEM with an explicit formula for the stability parameter τ_K that incorporates the flow direction, has the capability to deal with problems where there is substantial variation of the Péclet number, and gives the same limit as the RFB method. The method produces the same a priori error estimates that are typically obtained with streamline‐upwind Petrov/Galerkin and RFB. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Convergence phenomena of \documentclass{article} \usepackage{amsmath,amsfonts}\pagestyle{empty}\begin{document} $\mathcal{Q}_p$ \end{document}‐elements for convection–diffusion problems

We present a numerical study for singularly perturbed convection–diffusion problems using higher order Galerkin and streamline diffusion finite element method. We are especially interested in convergence and superconvergence properties with respect to different interpolation operators. For this we investigate pointwise interpolation and vertex‐edge‐cell interpolation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Construction of locally conservative fluxes for the SUPG method

We consider the construction of locally conservative fluxes by means of a simple postprocessing technique obtained from the finite element solutions of advection diffusion equations. It is known that a naive calculation of fluxes from these solutions yields nonconservative fluxes. We consider two finite element methods: the usual continuous Galerkin finite element method for solving nondominating advection diffusion equations and the streamline upwind/Petrov‐Galerkin method for solving advection dominated problems. We then describe the postprocessing technique for constructing conservative fluxes from the numerical solutions of the general variational formulation. The postprocessing technique requires solving an auxiliary Neumann boundary value problem on each element independently and it produces a locally conservative flux on a vertex centered dual mesh relative to the finite element mesh. We provide a convergence analysis for the postprocessing technique. Performance of the technique and the convergence behavior are demonstrated through numerical examples including a set of test problems for advection diffusion equations, advection dominated equations, and drift‐diffusion equations. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1971–1994, 2015     

输运方程谱间断流线扩散耦合方法

本文讨论了谱有限元方法,构造了求解Ｂｏｌｔｚｍ ａｎｎ 方程球谐函数谱展开和间断流线扩散有限元耦合格式．建立了这种耦合方法的稳定性及最优阶收敛性误差估计．得到了比标准有限元更高的精度．     

中子测井问题中非定常输运方程的谱流线扩散有限元耦合方法

针对中子测井问题,研究非定常Boltamann中子输运方程的确定型数值求解方法,给出了求解Boltzmann方程的球谐函数展开和流线扩散有限元耦合方法,证明了这种耦合方法的收敛性和误差估计。实际算例表明此方法是有效的。     

对流扩散方程的稳定化流线扩散法最小二乘非协调有限元分析

将最小二乘法和稳定化的流线扩散法相结合,研究了对流扩散方程的非协调有限元格式,用矩形EQ_1~(rot)元和零阶R-T元分别来逼近位移和应力,利用单元本身的特殊性质,证明了离散格式解的存在惟一性,得到了位移H~1-模和应力H(div)-模的最优误差估计.     

不可压混溶驱替问题的流线扩散──混合元数值模拟

采用标准元模拟不可压混溶流问题，当扩散系数矩阵小过剖分参数时，有限元格式仅能给出比最优精度低一阶的逼近解，格式稳定性差并伴有强烈的数值弥散现象．为了克服上述缺陷，本文对压力方程采用混合元，而对浓度方程采用流线扩散格式，在扩散矩阵为线性的假定下，证明了该格式具有较标准元更高的逼近精度（比最优阶低1／2）和更好的稳定性．     

A study of isogeometric analysis for scalar convection–diffusion equations

Isogeometric analysis (IGA), in combination with the streamline upwind Petrov–Galerkin (SUPG) stabilization, is studied for the discretization of steady-state convection–diffusion equations. Numerical results obtained for the Hemker problem are compared with results computed with the SUPG finite element method of the same order. Using an appropriate parameterization for IGA, the computed solutions are much more accurate than those obtained with the finite element method, both in terms of the size of spurious oscillations and of the sharpness of layers.     

非定常线性化Navier-Stokes方程的非协调流线扩散有限元法分析

对非定常线性化Navier-Stokes方程提出了非协调流线扩散有限元方法．用向后Euler格式离散时间,用流线扩散法处理扩散项带来的非稳定性．速度采用不连续的分片线性逼近,压力采用分片常数逼近．得到了离散解的存在唯一性以及在一定范数意义下离散解的稳定性和误差估计．     

A HIGH ORDER ADAPTIVE FINITE ELEMENT METHOD FOR SOLVING NONLINEAR HYPERBOLIC CONSERVATION LAWS

In this note,we apply the h-adaptive streamline diffusion finite element method with a small mesh-dependent artificial viscosity to solve nonlinear hyperbolic p...     

The characteristic finite difference streamline diffusion method for convection-dominated diffusion problems

In this paper, we consider the characteristic finite difference streamline diffusion method for two-dimensional convection-dominated diffusion problems. The scheme is combined the method of characteristics with the finite difference streamline diffusion (FDSD) method to create the characteristic FDSD (C-FDSD) procedures. Stability analysis and error estimate of the C-FDSD method are deduced. The scheme not only realizes the purpose of lowering the time-truncation error, using larger time step for solving the convection-dominated diffusion problems, but also keeps the favorable stability and high precision of the FDSD method. Finally, numerical experiments are presented to illustrate the availability of the scheme.     

Pointwise error estimates of the bilinear SDFEM on Shishkin meshes

A model singularly perturbed convection–diffusion problem in two space dimensions is considered. The problem is solved by a streamline diffusion finite element method (SDFEM) that uses piecewise bilinear finite elements on a Shishkin mesh. We prove that the method is convergent, independently of the diffusion parameter ε, with a pointwise accuracy of almost order 11/8 outside and inside the boundary layers. Numerical experiments support these theoretical results. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

三维热传导型半导体器件瞬态模拟问题的Crank-Nicolson差分-流线扩散有限元法及其数值分析

本文研究三维热传导型半导体器件瞬态模拟问题的数值方法.针对数学模型中各方程不同的特点,分别提出不同的有限元格式.特别针对浓度方程组是对流为主扩散问题的特点,使用Crank-Nicolson差分-流线扩散计算格式,提高了数值解的稳定性.得到的L2误差估计关于空间剖分步长是拟最优的,关于时间步长具有二阶精度.     

A NEW STABILIZED FINITE ELEMENT METHOD FOR SOLVING TRANSIENT NAVIER-STOKES EQUATIONS WITH HIGH REYNOLDS NUMBER

In this paper, we present a new stabilized finite element method for transient Navier-Stokes equations with high Reynolds number based on the projection of the velocity and pressure. We use Taylor-Hood elements and the equal order elements in space and second order difference in time to get the fully discrete scheme. The scheme is proven to possess the absolute stability and the optimal error estimates. Numerical experiments show that our method is effective for transient Navier-Stokes equations with high Reynolds number and the results are in good agreement with the value of subgrid-scale eddy viscosity methods, Petro-Galerkin finite element method and streamline diffusion method.     

SDFEM with non-standard higher-order finite elements for a convection-diffusion problem with characteristic boundary layers

Considering a singularly perturbed problem with exponential and characteristic layers, we show convergence for non-standard higher-order finite elements using the streamline diffusion finite element method (SDFEM). Moreover, for the standard higher-order space Q_p\mathcal {Q}_{p} supercloseness of the numerical solution w.r.t. an interpolation of the exact solution in the streamline diffusion norm of order p+1/2 is proved.     

Supercloseness of the SDFEM on Shishkin triangular meshes for problems with exponential layers

In this paper, we analyze the supercloseness property of the streamline diffusion finite element method (SDFEM) on Shishkin triangular meshes, which is different from one in the case of rectangular meshes. The analysis depends on integral inequalities for the parts related to the diffusion in the bilinear form. Moreover, our result allows the construction of a simple postprocessing that yields a more accurate solution. Finally, numerical experiments support these theoretical results.     

Pointwise estimates of SDFEM on Shishkin triangular meshes for problems with characteristic layers

In this paper, we present pointwise estimates of the streamline diffusion finite element method (SDFEM) for conforming piecewise linears on Shishkin triangular meshes. The method is applied to a model singularly perturbed convection-diffusion problem with characteristic layers. Using a new variant of artificial crosswind diffusion, we prove that uniformly pointwise error bounds away from the layers are of order almost 7/4 (up to a logarithmic factor). In some cases, the convergence order is almost 15/8. Our analysis depends on discrete Green’s functions and sharp estimates of the diffusion and convection parts in the bilinear form. Finally, numerical experiments support our theoretical results.     

A modified streamline diffusion method for solving the stationary Navier-Stokes equation

Summary Recently, Hughes et al. [11, 12] proposed new finite element schemes of Petrov-Galerkin type for solving the Stokes problem which do not require the discrete version of the Ladyshenskaya-Babuka-Brezzi-condition (LBB-condition). In this paper we derive a conforming finite element method for solving the stationary Navier-Stokes equations which combines the advantages of arbitrary finite element spaces for velocity/pressure with the favourable properties of the streamline diffusion method in the case of moderate and high Reynolds number.     

多孔介质中两相不可压混溶驱替的差分流线扩散法及其误差分析

多孔介质中两相不可压混熔驱替问题可描述为椭圆和抛物耦合的非线性偏微分方程组，对椭圆方程采用混合元方法，而对抛物方程采用差分流线扩散法，本文构造了求解该问题的差分流线扩散-混合元格式，最后，给出所构造格式按L^∞(L^2）模的拟最优误差阶估计。     

A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

In this note, a non‐standard finite difference (NSFD) scheme is proposed for an advection‐diffusion‐reaction equation with nonlinear reaction term. We first study the diffusion‐free case of this equation, that is, an advection‐reaction equation. Two exact finite difference schemes are constructed for the advection‐reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection‐reaction equation is combined with a finite difference space‐approximation of the diffusion term to provide a NSFD scheme for the advection‐diffusion‐reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd.