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-Level Finite Element Iterative Algorithm Based on Stabilized Method for the Stationary Incompressible Magnetohydrodynamics

In this paper, based on the stabilization technique, the Oseen iterative method and the two-level finite element algorithm are combined to numerically solve the stationary incompressible magnetohydrodynamic (MHD) equations. For the low regularity of the magnetic field, when dealing with the magnetic field sub-problem, the Lagrange multiplier technique is used. The stabilized method is applied to approximate the flow field sub-problem to circumvent the inf-sup condition restrictions. One- and two-level stabilized finite element algorithms are presented, and their stability and convergence analysis is given. The two-level method uses the Oseen iteration to solve the nonlinear MHD equations on a coarse grid of size H, and then employs the linearized correction on a fine grid with grid size h. The error analysis shows that when the grid sizes satisfy h=O(H2), the two-level stabilization method has the same convergence order as the one-level one. However, the former saves more computational cost than the latter one. Finally, through some numerical experiments, it has been verified that our proposed method is effective. The two-level stabilized method takes less than half the time of the one-level one when using the second class Nédélec element to approximate magnetic field, and even takes almost a third of the computing time of the one-level one when adopting the first class Nédélec element.     

Numerical Analysis and Comparison of Three Iterative Methods Based on Finite Element for the 2D/3D Stationary Micropolar Fluid Equations

In this paper, three iterative methods (Stokes, Newton and Oseen iterative methods) based on finite element discretization for the stationary micropolar fluid equations are proposed, analyzed and compared. The stability and error estimation for the Stokes and Newton iterative methods are obtained under the strong uniqueness conditions. In addition, the stability and error estimation for the Oseen iterative method are derived under the uniqueness condition of the weak solution. Finally, numerical examples test the applicability and the effectiveness of the three iterative methods.     

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.     

Finite Element Iterative Methods for the 3D Steady Navier–Stokes Equations

In this work, a finite element (FE) method is discussed for the 3D steady Navier–Stokes equations by using the finite element pair Xh×Mh. The method consists of transmitting the finite element solution (uh,ph) of the 3D steady Navier–Stokes equations into the finite element solution pairs (uhn,phn) based on the finite element space pair Xh×Mh of the 3D steady linearized Navier–Stokes equations by using the Stokes, Newton and Oseen iterative methods, where the finite element space pair Xh×Mh satisfies the discrete inf-sup condition in a 3D domain Ω. Here, we present the weak formulations of the FE method for solving the 3D steady Stokes, Newton and Oseen iterative equations, provide the existence and uniqueness of the FE solution (uhn,phn) of the 3D steady Stokes, Newton and Oseen iterative equations, and deduce the convergence with respect to (σ,h) of the FE solution (uhn,phn) to the exact solution (u,p) of the 3D steady Navier–Stokes equations in the H1−L2 norm. Finally, we also give the convergence order with respect to (σ,h) of the FE velocity uhn to the exact velocity u of the 3D steady Navier–Stokes equations in the L2 norm.     

Error Analysis and Adaptive Methods of Least Squares Nonconforming Finite Element for the Transport Equations

In this paper, we consider a least squares nonconforming finite element of low order for solving the transport equations. We give a detailed overview on the stability and the convergence properties of our considered methods in the stability norm. Moreover, we derive residual type a posteriori error estimates for the least squares nonconforming finite element methods under $H^{−1}$-norm, which can be used as the error indicators to guide the mesh refinement procedure in the adaptive finite element method. The theoretical results are supported by a series of numerical experiments.     

A New Finite Volume Element Formulation for the Non-Stationary Navier-Stokes Equations

A semi-discrete scheme about time for the non-stationary Navier-Stokes equations is presented firstly, then a new fully discrete finite volume element (FVE) formulation based on macroelement is directly established from the semi-discrete scheme about time. And the error estimates for the fully discrete FVE solutions are derived by means of the technique of the standard finite element method. It is shown by numerical experiments that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary Navier-Stokes equations and it is one of the most effective numerical methods among the FVE formulation, the finite element formulation, and the finite difference scheme.     

An Error Analysis for the Finite Element Approximation to the Steady-State Poisson-Nernst-Planck Equations

Poisson-Nernst-Planck equations are a coupled system of nonlinear partial differential equations consisting of the Nernst-Planck equation and the electrostatic Poisson equation with delta distribution sources, which describe the electrodiffusion of ions in a solvated biomolecular system. In this paper, some error bounds for a piecewise finite element approximation to this problem are derived. Several numerical examples including biomolecular problems are shown to support our analysis.     

A Stabilized Finite Element Method for Non-Stationary Conduction-Convection Problems

This paper is concerned with a stabilized finite element method based on two local Gauss integrations for the two-dimensional non-stationary conduction-convection equations by using the lowest equal-order pairs of finite elements. This method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf-sup condition. The stability of the discrete scheme is derived under some regularity assumptions. Optimal error estimates are obtained by applying the standard Galerkin techniques. Finally, the numerical illustrations agree completely with the theoretical expectations.     

Uniform Finite Element Error Estimates with Power-Type Asymptotic Constants for Unsteady Navier–Stokes Equations

Uniform error estimates with power-type asymptotic constants of the finite element method for the unsteady Navier–Stokes equations are deduced in this paper. By introducing an iterative scheme and studying its convergence, we firstly derive that the solution of the Navier–Stokes equations is bounded by power-type constants, where we avoid applying the Gronwall lemma, which generates exponential-type factors. Then, the technique is extended to the error estimate of the long-time finite element approximation. The analyses show that, under some assumptions on the given data, the asymptotic constants in the finite element error estimates for the unsteady Navier–Stokes equations are uniformly power functions with respect to the initial data, the viscosity, and the body force for all time t>0. Finally, some numerical examples are shown to verify the theoretical predictions.     

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.     

Finite Element Analysis of Maxwell's Equations in Dispersive Lossy Bi-Isotropic Media

In this paper, the time-dependent Maxwell's equations used to modeling wave propagation in dispersive lossy bi-isotropic media are investigated. Existence and uniqueness of the modeling equations are proved. Two fully discrete finite element schemes are proposed, and their practical implementation and stability are discussed.     

Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods

In this paper, we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method. We use two Newton iterations on the fine grid in our methods. Firstly, we solve an original nonlinear problem on the coarse nonlinear grid, then we use Newton iterations on the fine grid twice. The two-grid idea is from Xu$'$s work [SIAM J. Numer. Anal., 33 (1996), pp. 1759-1777] on standard finite method. We also obtain the error estimates for the algorithms of the two-grid method. It is shown that the algorithm achieves asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy $h=\mathcal{O}(H^{(4k+1)/(k+1)})$.     

A Finite Volume Method Based on the Constrained Nonconforming Rotated Q1-Constant Element for the Stokes Problem

We construct a finite volume element method based on the constrained nonconforming rotated Q₁-constant element (CNRQ₁-P₀) for the Stokes problem. Two meshes are needed, which are the primal mesh and the dual mesh. We approximate the velocity by CNRQ₁ elements and the pressure by piecewise constants. The errors for the velocity in the H¹ norm and for the pressure in the L² norm are O(h) and the error for the velocity in the L² norm is O(h²). Numerical experiments are presented to support our theoretical results.     

缩减基有限元方法快速求解参数化偏微分方程

针对可用参数化微分方程表示的问题,缩减基有限元方法利用典型参数的高精度解构造基函数,可大幅度降低待求问题的自由度.以固体热传导和中子扩散的快速求解为例,展示该方法的优良特性.结果表明：在线阶段的求解效率可以实现两到三个数量级的提升.基于高保真模拟的缩减基模型可使高性能计算有效地应用于工程优化设计、应急指挥及复杂问题的反分析等.     

Error Estimates and Superconvergence of Mixed Finite Element Methods for Optimal Control Problems with Low Regularity

In this paper, we investigate the error estimates and superconvergence property of mixed finite element methods for elliptic optimal control problems. 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 $L^\infty$-error estimates for the control variable. Moreover, using a recovery operator, we also derive some superconvergence results for the control variable. Finally, a numerical example is given to demonstrate the theoretical results.     

非定常Navier-Stokes方程基于两重网格离散的有限元并行算法

基于两重网格离散和区域分解技巧,提出三种求解非定常Navier-Stokes方程的有限元并行算法.算法的基本思想是在每一时间迭代步,在粗网格上采用Oseen迭代法求解非线性问题,在细网格上分别并行求解Oseen、Newton、Stokes线性问题以校正粗网格解.对于空间变量采用有限元离散,时间变量采用向后Euler格式离散.数值实验验证了算法的有效性.     

流体力学方程的间断有限元方法

在二维区域三角形网格上应用一阶、二阶和三阶精度间断有限元方法,对流体力学方程和方程组进行了数值模拟.计算结果与差分方法计算结果比较,认为间断有限元方法在求解复杂边界条件和区域问题上有一定的优势.     

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。     

Lagrange坐标系下二维气动方程组的RKDG有限元方法

构造Lagrange坐标系下二维可压缩气动方程组的RKDG(Runge-Kutta Discontinuous Galerkin)有限元方法.将流体力学方程组和几何守恒律统-求解,所有计算都在固定的网格上进行,计算过程中不需要网格节点的速度信息.对几个数值算例进行数值模拟,得到较好的数值模拟结果.