Two-Level Defect-Correction Method for Steady Navier-Stokes Problem with Friction Boundary Conditions

In this paper, we present two-level defect-correction finite element method for steady Navier-Stokes equations at high Reynolds number with the friction boundary conditions, which results in a variational inequality problem of the second kind. Based on Taylor-Hood element, we solve a variational inequality problem of Navier-Stokes type on the coarse mesh and solve a variational inequality problem of Navier-Stokes type corresponding to Newton linearization on the fine mesh. The error estimates for the velocity in the $H^1$ norm and the pressure in the $L^2$ norm are derived. Finally, the numerical results are provided to confirm our theoretical analysis.     

Two-Level Stabilized Finite Volume Methods for the Stationary Navier-Stokes Equations

In this work, two-level stabilized finite volume formulations for the 2D steady Navier-Stokes equations are considered. These methods are based on the local Gauss integration technique and the lowest equal-order finite element pair. Moreover, the two-level stabilized finite volume methods involve solving one small Navier-Stokes problem on a coarse mesh with mesh size $H$, a large general Stokes problem for the Simple and Oseen two-level stabilized finite volume methods on the fine mesh with mesh size $h$=$\mathcal{O}(H^2)$ or a large general Stokes equations for the Newton two-level stabilized finite volume method on a fine mesh with mesh size $h$=$\mathcal{O}(|\log h|^{1/2}H^3)$. These methods we studied provide an approximate solution $(\widetilde{u}_h^v,\widetilde{p}_h^v)$ with the convergence rate of same order as the standard stabilized finite volume method, which involve solving one large nonlinear problem on a fine mesh with mesh size $h$. Hence, our methods can save a large amount of computational time.     

Two-Scale Picard Stabilized Finite Volume Method for the Incompressible Flow

In this paper, we consider a two-scale stabilized finite volume method for the two-dimensional stationary incompressible flow approximated by the lowest equal-order element pair $P_1-P_1$ which does not satisfy the inf-sup condition. The two-scale method consists of solving a small non-linear system on the coarse mesh and then solving a linear Stokes equations on the fine mesh. Convergence of the optimal order in the $H^1$-norm for velocity and the $L^2$-norm for pressure is obtained. The error analysis shows there is the same convergence rate between the two-scale stabilized finite volume solution and the usual stabilized finite volume solution on a fine mesh with relation $h =\mathcal{O}(H^2)$. Numerical experiments completely confirm theoretic results. Therefore, this method presented in this paper is of practical importance in scientific computation.     

Finite Element $θ$-Schemes for the Acoustic Wave Equation

In this paper, we investigate the stability and convergence of a family of implicit finite difference schemes in time and Galerkin finite element methods in space for the numerical solution of the acoustic wave equation. The schemes cover the classical explicit second-order leapfrog scheme and the fourth-order accurate scheme in time obtained by the modified equation method. We derive general stability conditions for the family of implicit schemes covering some well-known CFL conditions. Optimal error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the $L^2$-norm error over a finite time interval converges optimally as $\mathcal{O}(h^{p+1}+∆t^s)$, where $p$ denotes the polynomial degree, $s$=2 or 4, $h$ the mesh size, and $∆t$ the time step.     

大雷诺数Navier-Stokes方程的两水平亚格子模型稳定化方法

基于两重网格离散方法,提出三种求解大雷诺数定常Navier-Stokes方程的两水平亚格子模型稳定化有限元算法.其基本思想是首先在一粗网格上求解带有亚格子模型稳定项的Navier-Stokes方程,然后在细网格上分别用三种不同的校正格式求解一个亚格子模型稳定化的线性问题,以校正粗网格解.通过适当的稳定化参数和粗细网格尺寸的选取,这些算法能取得最优渐近收敛阶的有限元解.最后,用数值模拟验证三种算法的有效性.     

A Priori Error Estimates of Crank-Nicolson Finite Volume Element Method for Parabolic Optimal Control Problems

In this paper, the Crank-Nicolson linear finite volume element method is applied to solve the distributed optimal control problems governed by a parabolic equation. The optimal convergent order $\mathcal{O}(h^2+k^2)$ is obtained for the numerical solution in a discrete $L^2$-norm. A numerical experiment is presented to test the theoretical result.     

Two-Grid Discretization Scheme for Nonlinear Reaction Diffusion Equation by Mixed Finite Element Methods

In this paper, we study an efficient scheme for nonlinear reaction-diffusion equations discretized by mixed finite element methods. We mainly concern the case when pressure coefficients and source terms are nonlinear. To linearize the nonlinear mixed equations, we use the two-grid algorithm. We first solve the nonlinear equations on the coarse grid, then, on the fine mesh, we solve a linearized problem using Newton iteration once. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy $H =\mathcal{O}(h^{\frac{1}{2}})$. As a result, solving such a large class of nonlinear equations will not be much more difficult than getting solutions of one linearized system.     

不可压缩流的并行两水平稳定有限元算法

提出一种数值求解定常不可压缩Stokes方程的并行两水平Grad-div稳定有限元算法。首先在粗网格中求解Grad-div稳定化的全局解, 再在相互重叠的细网格子区域上并行纠正。通过对稳定化参数、粗细网格尺寸恰当的选取, 该方法可得到最优收敛率, 数值结果验证了算法的高效性。     

Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions

The theory of a class of spectral methods is extended to Volterra integro-differential equations which contain a weakly singular kernel $(t-s)^{-\mu}$ with $0<\mu<1$. In this work, we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth. We provide a rigorous error analysis for the spectral methods, which shows that both the errors of approximate solutions and the errors of approximate derivatives of the solutions decay exponentially in $L^\infty$-norm and weighted $L^2$-norm. The numerical examples are given to illustrate the theoretical results.     

An effective explicit pressure gradient scheme implemented in the two-level non-staggered grids for incompressible Navier–Stokes equations

In this paper, an improved two-level method is presented for effectively solving the incompressible Navier–Stokes equations. This proposed method solves a smaller system of nonlinear Navier–Stokes equations on the coarse mesh and needs to solve the Oseen-type linearized equations of motion only once on the fine mesh level. Within the proposed two-level framework, a prolongation operator, which is required to linearize the convective terms at the fine mesh level using the convergent Navier–Stokes solutions computed at the coarse mesh level, is rigorously derived to increase the prediction accuracy. This indispensable prolongation operator can properly communicate the flow velocities between the two mesh levels because it is locally analytic. Solution convergence can therefore be accelerated. For the sake of numerical accuracy, momentum equations are discretized by employing the general solution for the two-dimensional convection–diffusion–reaction model equation. The convective instability problem can be simultaneously eliminated thanks to the proper treatment of convective terms. The converged solution is, thus, very high in accuracy as well as in yielding a quadratic spatial rate of convergence. For the sake of programming simplicity and computational efficiency, pressure gradient terms are rigorously discretized within the explicit framework in the non-staggered grid system. The proposed analytical prolongation operator for the mapping of solutions from the coarse to fine meshes and the explicit pressure gradient discretization scheme, which accommodates the dispersion-relation-preserving property, have been both rigorously justified from the predicted Navier–Stokes solutions.     

A Maximum Entropy Method Based on Orthogonal Polynomials for Frobenius-Perron Operators

Let $S$: [0, 1]→[0, 1] be a chaotic map and let $f^∗$ be a stationary density of the Frobenius-Perron operator $P_S$: $L^1$→$L^1$ associated with $S$. We develop a numerical algorithm for approximating $f^∗$, using the maximum entropy approach to an under-determined moment problem and the Chebyshev polynomials for the stability consideration. Numerical experiments show considerable improvements to both the original maximum entropy method and the discrete maximum entropy method.     

An Algebraic Multigrid Method for Nearly Incompressible Elasticity Problems in Two-Dimensions

In this paper, we discuss an algebraic multigrid (AMG) method for nearly incompressible elasticity problems in two-dimensions. First, a two-level method is proposed by analyzing the relationship between the linear finite element space and the quartic finite element space. By choosing different smoothers, we obtain two types of two-level methods, namely TL-GS and TL-BGS. The theoretical analysis and numerical results show that the convergence rates of TL-GS and TL-BGS are independent of the mesh size and the Young's modulus, and the convergence of the latter is greatly improved on the order $p$. However, the convergence of both methods still depends on the Poisson's ratio. To fix this, we obtain a coarse level matrix with less rigidity based on selective reduced integration (SRI) method and get some types of two-level methods by combining different smoothers. With the existing AMG method used as a solver on the first coarse level, an AMG method can be finally obtained. Numerical results show that the resulting AMG method has better efficiency for nearly incompressible elasticity problems.     

A Two-Level Method for Pressure Projection Stabilized P1 Nonconforming Approximation of the Semi-Linear Elliptic Equations

In this paper, we study a new stabilized method based on the local pressure projection to solve the semi-linear elliptic equation. The proposed scheme combines nonconforming finite element pairs NCP₁−P₁triangle element and two-level method, which has a number of attractive computational properties: parameter-free, avoiding higher-order derivatives or edge-based data structures, but have more favorable stability and less support sets. Stability analysis and error estimates have been done. Finally, numerical experiments to check estimates are presented.     

Convergence Analysis of Legendre-Collocation Methods for Nonlinear Volterra Type Integro Equations

A Legendre-collocation method is proposed to solve the nonlinear Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors in $L^2$-norm and $L^\infty$-norm will decay exponentially provided that the kernel function is sufficiently smooth. Numerical results are presented, which confirm the theoretical prediction of the exponential rate of convergence.     

A Priori Error Estimates of Finite Element Methods for Linear Parabolic Integro-Differential Optimal Control Problems

In this paper, we study the mathematical formulation for an optimal control problem governed by a linear parabolic integro-differential equation and present the optimality conditions. We then set up its weak formulation and the finite element approximation scheme. Based on these we derive the a priori error estimates for its finite element approximation both in $H^1$ and $L^2$ norms. Furthermore, some numerical tests are presented to verify the theoretical results.     

叶栅全三维粘性反问题的数值解

本文发展了一种解叶栅全三维粘性反问题的新的数值方法.基于非正交曲线坐标与相应的非正交速度分量下完全守恒型的Navier－Stokes方程,全三维反问题规定叶片表面的无量纲压力分布反求叶型。计算中叶片表面的边界条件采用一种特殊的方式来处理，即一方面强加给定的压力分布条件,另方面叶面的几何位置在迭代过程中又是可移动的，其移动速度将与Navier—Stokes方程在当地的解联系起来，从而形成一种解定常问题的新的不定常过程.试算证明了本文方法的可行性。     

Error Estimates of Mixed Methods for Optimal Control Problems Governed by General Elliptic Equations

In this paper, we investigate the error estimates of mixed finite element methods for optimal control problems governed by general elliptic equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We derive $L^2$ and $H^{-1}$-error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.     

气体中心式离心喷嘴喷雾实验与三维仿真

在实验的基础上, 基于RNG k-ε模型对常压下气体中心式同轴离心(gas-centered swirl coaxial,GCSC)喷嘴喷雾形态和破碎模式进行了三维仿真研究。采用网格自适应加密(adaptive mesh refinement,AMR)技术、耦合水平集和流体体积(coupled level-set and volume of fluid, CLSVOF)方法对气液界面进行捕捉。结果表明, 液体质量流率($\dot{m}_{\mathrm{l}}$)不变, 随着气体质量流率($\dot{m}_{\mathrm{g}}$)的增加, 中心气流的引射作用增强, 液膜内外压差增大, 雾化锥角减小, 并对其流动特性进行了分析; 而$\dot{m}_{\mathrm{g}}$不变时, 液膜在喷嘴出口的径向速度与切向速度随$\dot{m}_{\mathrm{l}}$的增大而增大, 导致雾化锥角增大。同时根据气液质量流率比(gas-liquid mass flow rate,GLR), 将喷雾的破碎模式分为穿孔破碎、气泡破碎和气动破碎。      

An Iterative Two-Grid Method of a Finite Element PML Approximation for the Two Dimensional Maxwell Problem

In this paper, we propose an iterative two-grid method for the edge finite element discretizations (a saddle-point system) of Perfectly Matched Layer (PML) equations to the Maxwell scattering problem in two dimensions. Firstly, we use a fine space to solve a discrete saddle-point system of $H(grad)$ variational problems, denoted by auxiliary system 1. Secondly, we use a coarse space to solve the original saddle-point system. Then, we use a fine space again to solve a discrete$\boldsymbol{H}(curl)$-elliptic variational problems, denoted by auxiliary system 2. Furthermore, we develop a regularization diagonal block preconditioner for auxiliary system 1 and use $H$-$X$ preconditioner for auxiliary system 2. Hence we essentially transform the original problem in a fine space to a corresponding (but much smaller) problem on a coarse space, due to the fact that the above two preconditioners are efficient and stable. Compared with some existing iterative methods for solving saddle-point systems, such as PMinres, numerical experiments show the competitive performance of our iterative two-grid method.     

Spectral-Collocation Method for Volterra Delay Integro-Differential Equations with Weakly Singular Kernels

A spectral Jacobi-collocation approximation is proposed for Volterra delay integro-differential equations with weakly singular kernels. In this paper, we consider the special case that the underlying solutions of equations are sufficiently smooth. We provide a rigorous error analysis for the proposed method, which shows that both the errors of approximate solutions and the errors of approximate derivatives decay exponentially in $L^∞$ norm and weighted $L^2$ norm. Finally, two numerical examples are presented to demonstrate our error analysis.