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1.
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. 相似文献
2.
A Priori Error Estimates of Crank-Nicolson Finite Volume Element Method for Parabolic Optimal Control Problems
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Xianbing Luo Yanping Chen & Yunqing Huang 《advances in applied mathematics and mechanics.》2013,5(5):688-704
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. 相似文献
3.
This paper deals with the
two-level Newton iteration method based on the pressure projection
stabilized finite element approximation to solve the numerical solution of
the Navier-Stokes type variational inequality problem. We solve a small
Navier-Stokes problem on the coarse mesh with mesh size $H$ and solve a large linearized Navier-Stokes problem on the
fine mesh with mesh size $h$. The error estimates derived show that
if we choose $h=\mathcal{O}(|\log h|^{1/2}H^3)$, then the two-level method we
provide has the same $H^1$ and $L^2$ convergence orders of the velocity
and the pressure as the one-level stabilized
method. However, the $L^2$ convergence order of the velocity
is not consistent with that of one-level stabilized method.
Finally, we give the numerical results to
support the theoretical analysis. 相似文献
4.
Samir Karaa 《advances in applied mathematics and mechanics.》2011,3(2):181-203
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. 相似文献
5.
Error Analysis and Adaptive Methods of Least Squares Nonconforming Finite Element for the Transport Equations
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Huipo Liu Shuanghu Wang & Hongbin Han 《advances in applied mathematics and mechanics.》2016,8(5):871-886
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. 相似文献
6.
Two-Level Defect-Correction Method for Steady Navier-Stokes Problem with Friction Boundary Conditions
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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. 相似文献
7.
Two-Grid Discretization Scheme for Nonlinear Reaction Diffusion Equation by Mixed Finite Element Methods
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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. 相似文献
8.
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。 相似文献
9.
An Inf-Sup Stabilized Finite Element Method by Multiscale Functions for the Stokes Equations
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Zhihao Ge Yinnian He & Lingyu Song 《advances in applied mathematics and mechanics.》2009,1(2):273-287
In the paper, an inf-sup stabilized finite element method by multiscale
functions for the Stokes equations is discussed. The key idea is to use a Petrov-Galerkin approach based on the enrichment of the standard polynomial space for
the velocity component with multiscale functions. The inf-sup condition for $P_1$-$P_0$ triangular element (or $Q_1$-$P_0$ quadrilateral element) is established. And the optimal
error estimates of the stabilized finite element method for the Stokes equations
are obtained. 相似文献
10.
求解跨音速粘性流反问题的有限体积法 总被引:3,自引:0,他引:3
1前言叶轮机械气动热力学研究的最终目的之一,是为工程师提供可用以设计高性能、高效率叶轮机械的思想与方法。而在过去叶轮机械气动热力学的研究大都集中求解正问题上,即在给定几何形状的叶片的条件下来求解流场上的气动参数。因此,为了设计出高性能的叶型,通常是凭设计者的经验,通过试验或正问题流场的分析计算,对一系列几何形状相差不大的叶型进行筛选。用这种方法是非常耗时并导致昂贵的设计费用,而且这样也并不总能得到预期的效果。所以,正问题计算虽然在预测性能和筛选试验方案等方面具有重要价值,但对于叶栅设计,应该具有… 相似文献
11.
An Iterative Two-Grid Method of a Finite Element PML Approximation for the Two Dimensional Maxwell Problem
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Chunmei Liu Shi Shu Yunqing Huang Liuqiang Zhong & Junxian Wang 《advances in applied mathematics and mechanics.》2012,4(2):175-189
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. 相似文献
12.
13.
In this paper we present a process that includes both model/mesh repair and mesh generation. The repair algorithm is based on an initial mesh that may be either an initial mesh of a dirty CAD model or STL triangulation with many errors such as gaps, overlaps and T-junctions. This initial mesh is then remeshed by computing a discrete parametrization with Radial Basis Functions (RBF’s).We showed in [1] that a discrete parametrization can be computed by solving Partial Differential Equations (PDE’s) on an initial correct mesh using finite elements. Paradoxically, the meshless character of the RBF’s makes it an attractive numerical method for solving the PDE’s for the parametrization in the case where the initial mesh contains errors or holes. In this work, we implement the Orthogonal Gradients method to be described in [2], as a RBF solution method for solving PDE’s on arbitrary surfaces.Different examples show that the presented method is able to deal with errors such as gaps, overlaps, T-junctions and that the resulting meshes are of high quality. Moreover, the presented algorithm can be used as a hole-filling algorithm to repair meshes with undesirable holes. The overall procedure is implemented in the open-source mesh generator Gmsh [3]. 相似文献
14.
Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods
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Yanping Chen Peng Luan & Zuliang Lu 《advances in applied mathematics and mechanics.》2009,1(6):830-844
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
$h=\mathcal{O}(H^{(4k+1)/(k+1)})$. 相似文献
15.
A Spectral Method for Second Order Volterra Integro-Differential Equation with Pantograph Delay
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In this paper, a Legendre-collocation spectral method is developed for the
second order Volterra integro-differential equation with pantograph delay. We provide
a rigorous error analysis for the proposed method. The spectral rate of convergence
for the proposed method is established in both $L^2$-norm and $L^∞$-norm. 相似文献
16.
Yu Wang Songhe Song Zhijun Tan Desheng Wang 《Journal of computational physics》2009,228(17):6333-6348
This paper presents an adaptive method for variational curve smoothing based on level set implementation. A suitable cost functional is minimized via solving the derived Euler–Lagrangian equation, of which the discretization is conducted on unstructured triangular meshes by employing a simple and effective finite volume scheme. Through adaptive refinement of the mesh, the geometry features of the given curve can be well resolved in a cost-effective way. Various numerical experiments demonstrate the effectiveness and efficiency of the proposed approach. 相似文献
17.
18.
Several two-level iterative methods based on nonconforming finite element methods are applied for solving numerically the 2D/3D stationary incompressible MHD equations under different uniqueness conditions. These two-level algorithms are motivated by applying the m iterations on a coarse grid and correction once on a fine grid. A one-level Oseen iterative method on a fine mesh is further studied under a weak uniqueness condition. Moreover, the stability and error estimate are rigorously carried out, which prove that the proposed methods are stable and effective. Finally, some numerical examples corroborate the effectiveness of our theoretical analysis and the proposed methods. 相似文献
19.
Liping Liu Min Huang Kewei Yuan & Michal K?í ?ek 《advances in applied mathematics and mechanics.》2009,1(1):125-139
In this paper, we are concerned with the numerical approximation of
a steady-state heat radiation problem with a nonlinear Stefan-Boltzmann boundary
condition in$\mathbb{R}^3$. We first derive an equivalent minimization problem and then
present a finite element analysis to the solution of such a minimization problem.
Moreover, we apply the Newton iterative method for solving the nonlinear equation
resulting from the minimization problem. A numerical example is given to
illustrate theoretical results. 相似文献
20.
Shuhua Zhang & Jing Wang 《advances in applied mathematics and mechanics.》2016,8(5):827-846
In this paper, we study carbon emission trading whose market is gaining popularity as a policy instrument for global climate change. The mathematical model
is presented for pricing options on $CO_2$ emission allowance futures with jump diffusion
processes, and a so-called fitted finite volume method is proposed to solve the
pricing model for the spatial discretization, in which the Crank-Nicolson is employed
for time stepping. In addition, the stability and the convergence of the fully discrete
scheme are given, and some numerical results, which are compared with the closed
form solution and the Monte Carlo simulation solution, are provided to demonstrate
the rates of convergence and the robustness of the numerical method. 相似文献