Local and parallel finite element algorithms for time-dependent convection-diffusion equations

Local and parallel finite element algorithms based on two-grid discretization for the time-dependent convection-diffusion equations are presented. These algorithms are motivated by the observation that, for a solution to the convection-diffusion problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel proce- dures. Hence, these local and parallel algorithms only involve one small original problem on the coarse mesh and some correction problems on the local fine grid. One technical tool for the analysis is the local a priori estimates that are also obtained. Some numerical examples are given to support our theoretical analvsis.     

A parallel two-level finite element method for the Navier-Stokes equations

Based on domain decomposition, a parallel two-level finite element method for the stationary Navier-Stokes equations is proposed and analyzed. The basic idea of the method is first to solve the Navier-Stokes equations on a coarse grid, then to solve the resulted residual equations in parallel on a fine grid. This method has low communication complexity. It can be implemented easily. By local a priori error estimate for finite element discretizations, error bounds of the approximate solution are derived. Numerical results are also given to illustrate the high efficiency of the method.     

Local projection stabilized finite element method for Navier-Stokes equations

This paper extends the results of Matthies, Skrzypacz, and Tubiska for the Oseen problem to the Navier-Stokes problem. For the stationary incompressible Navier- Stokes equations, a local projection stabilized finite element scheme is proposed. The scheme overcomes convection domination and improves the restrictive inf-sup condition. It not only is a two-level approach but also is adaptive for pairs of spaces defined on the same mesh. Using the approximation and projection spaces defined on the same mesh, the scheme leads to much more compact stencils than other two-level approaches. On the same mesh, besides the class of local projection stabilization by enriching the approximation spaces, two new classes of local projection stabilization of the approximation spaces are derived, which do not need to be enriched by bubble functions. Based on a special interpolation, the stability and optimal prior error estimates are shown. Numerical results agree with some benchmark solutions and theoretical analysis very well.     

A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations

A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming （P1）2 - P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.     

A high-order characteristics/finite element method for the incompressible Navier-Stokes equations

In this paper we consider a discretization of the incompressible Navier-Stokes equations involving a second-order time scheme based on the characteristics method and a spatial discretization of finite element type. Theoretical and numerical analyses are detailed and we obtain stability results abnd optimal eror estimates on the velocity and pressure under a time step restriction less stringent than the standard Courant-Freidrichs-Levy condition. Finally, some numerical results obtained wiht the code N3S are shown which justify the interest of this scheme and its advantages with respect to an analogous first-order time scheme. © 1997 John Wiley & Sons, Ltd.     

Discontinuous element pressure gradient stabilizations for compressible Navier-Stokes equations based on local projections

A pressure gradient discontinuous finite element formulation for the compressible Navier-Stokes equations is derived based on local projections. The resulting finite element formulation is stable and uniquely solvable without requiring a B-B stability condition. An error estimate is Obtained.     

Parallel finite element algorithm based on full domain partition for stationary Stokes equations

Based on the full domain partition, a parallel finite element algorithm for the stationary Stokes equations is proposed and analyzed. In this algorithm, each subproblem is defined in the entire domain. Majority of the degrees of freedom are associated with the relevant subdomain. Therefore, it can be solved in parallel with other subproblems using an existing sequential solver without extensive recoding. This allows the algorithm to be implemented easily with low communication costs. Numerical results are given showing the high efficiency of the parallel algorithm.     

Residual a posteriori error estimate two-grid methods for the steady Navier-Stokes equation with stream function form

Residual based on a posteriori error estimates for conforming element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.     

Nonlinear Galerkin method for the exterior nonstationary Navier-Stokes equations

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A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations

We note in this study that the Navier-Stokes equations, when expressed in streamfunction-vorticity form, can be approximated to fourth-order accuracy with stencils extending only over a 3 x 3 square of points. The key advantage of the new compact fourth-order scheme is that it allows direct iteration for low-to-medium Reynolds numbers. Numerical solutions are obtained for the model problem of the driven cavity and compared with solutions available in the literature. For Re ? 7500 point-SOR iteration is used and the convergence is fast.     

基于CUDA的有限元矩阵并行装配算法研究

构建航天飞行器的结构有限元模型是准确模拟飞行仿真、完成飞行器在轨飞行阶段结构故障监测和诊断的基础。采用细长体飞行器简化梁模型，提出新的基于CUDA（Compute Unified Device Architecture）的有限元单元刚度矩阵生成和总刚度矩阵组装算法。依据梁单元矩阵的对称性，结合GPU硬件架构提出并行生成算法并进行改进。为有效减少装配时间，在装配过程中采用着色算法，提出了基于GPU（Graphics Processing Unit）共享内存的非零项组装策略，通过在不同计算平台下算例对比，验证了新算法的快速性。数值算例表明，本文算法的求解效率较高，针对一定计算规模内的模型可满足快速计算与诊断的实时性要求。     

TWO-GRID ERROR ESTIMATES FOR THE STREAM FUNCTION FORM OF NAVIER-STOKES EQUATIONS

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Low order nonconforming mixed finite element method for nonstationary incompressible Navier-Stokes equations

This paper studies a low order mixed finite element method (FEM) for nonstationary incompressible Navier-Stokes equations. The velocity and pressure are approximated by the nonconforming constrained Q ₁ ^rot element and the piecewise constant, respectively. The superconvergent error estimates of the velocity in the broken H ¹-norm and the pressure in the L ²-norm are obtained respectively when the exact solutions are reasonably smooth. A numerical experiment is carried out to confirm the theoretical results.     

Mixed finite element method of hexahedral elements for Navier-Stokes problem

In this paper, we derive a new mixed element format of hexahedral elements for Navier-Stokes problem in three-dimensional space.     

基于PVM的网络并行有限元面向对象的设计

子结构是有限元并行计算常用的一种方法,本文采用面向对象的方法,首先对子结构进行了面向对象的设计,得到了其类层次结构图;然后针对工作站网络有限元并行计算环境。提出了基于PVM消息传递平台上的Shadow—Mirror数据传输模型,该模型在有限元并行计算数据传输时,充分发挥数据面向对象的特性,采用设置数据缓冲区、短消息合并等方法以缩短数据通信时间,并据此编制了相应的程序。计算结果表明,使用文中提出的面向对象的Shadow—Mirror数据传输模型可以得到较为理想的并行加速比,而且随着问题规模增大,并行加速比增高。本文研究内容为进一步开展基于工作站网络的并行有限元研究提供了一个可参考的基础。     

二维定常不可压缩粘性流动N-S方程的数值流形方法

将流形方法应用于定常不可压缩粘性流动N-S方程的直接数值求解,建立基于Galerkin加权余量法的N-S方程数值流形格式,有限覆盖系统采用混合覆盖形式,即速度分量取1阶和压力取0阶多项式覆盖函数,非线性流形方程组采用直接线性化交替迭代方法和Nowton-Raphson迭代方法进行求解.将混合覆盖的四节点矩形流形单元用于阶梯流和方腔驱动流动的数值算例,以较少单元获得的数值解与经典数值解十分吻合.数值实验证明,流形方法是求解定常不可压缩粘性流动N-S方程有效的高精度数值方法.     

A flux-vector splitting method for steady Navier-Stokes equations

The flux-vector splitting method is applied to the convective part of the steady Navier-Stokes equations for incompressible flow. By the use of partial upwind differences in the split first-order part and central differences in the second-order part, a set of discrete equations is obtained which can be solved by vector variants of classical relaxation schemes. It is shown that accurate results can be obtained on one of the GAMM backward-facing step test problems.     

Finite element analysis of the steady Navier-Stokes equations by a multiplier method

In this paper a fully explicit finite element method (FEFEM) is presented for solving steady incompressible viscous flow problems. This full explicitness is achieved by combining the multiplier (or augmented Lagrangian) method with a pseudo-time-iteration method. FEFEM needs no global matrix at all and is of great advantage to large-scale problems because they can be solved within the limit of core memory. The optimum choice of a time increment and a penalty parameter is discussed and the driven cavity flow at a Reynolds number of 1000 is computed with a refined mesh (60 × 60 elements).     

Mixed time discontinuous space-time finite element method for convection diffusion equations

A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order equation is discretized with a space-time finite element method, continuous in space but discontinuous in time. Stability, existence, uniqueness and convergence of the approximate solutions are proved. Numerical results are presented to illustrate efficiency of the proposed method.     

A Galerkin finite element algorithm based on third-order Runge-Kutta temporal discretization along the uniform streamline for unsteady incompressible flows

In this paper, for two-dimensional unsteady incompressible flow, the Navier-Stokes equations without convection term are derived by the coordinate transformation along the streamline characteristic. The third-order Runge-Kutta method along the streamline is introduced to discrete the alternative Navier-Stokes equations in time, and spacial discretization is carried out by the Galerkin method, and then, the third-order accuracy finite element method is obtained. Meanwhile, the streamline velocity is uniformly approximated by initial velocity in each time step in order to reduce update frequency of total element matrix and improve calculation efficiency. Finally, some classic unsteady flow examples are calculated and analyzed by different calculation methods, which further demonstrate that the present method has more advantages in stability, permissible time step, dissipation, computational cost, and accuracy. The code can be downloaded at https://doi.org/10.13140/RG.2.2.27706.44484 .