定常的热传导-对流问题的非线性Galerkin/Petrov最小二乘混合元法

In this paper,a nonlinear Galerkin/Petrov-least squares mixed element (NG-PLSME) method for the stationary conduction-convection problems is presented and analyzed.The method is consistent and stable for any combination of dis-crete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition.The existence, uniqueness and convergence (at optimal rate) of the NGPLSME solution is proved in the case of sufficient viscosity (or small data).     

定常的热传导-对流问题的Galerkin/Petrov最小二乘混合元方法

1.引言 热传导-对流问题是大气动力学中的一个重要的方程,这个方程组也称为强迫耗散的非线性系统方程组,其较Navier-Stokes方程多了一个未知函数温度场,且温度与速度和压力之间存在着复杂的非线性关系.从热动力学可知,任何运动都会产生热量即有温度,而且温度与速度和压力之间必定互相转化,因此对该非线性系统的研究更具有实际意义.[1]先对     

热传导对流问题的自适应最小二乘Galerkin/Petrov混合有限元法

对热传导对流问题提出了自适应Galerkin/Petrov最小二乘混合有限元法．该算法对任何速度和压力有限元空间的组合是相容和稳定的(不需要满足Babuka-Brezzi稳定性条件)．利用Verfürth的一般理论,得到了热传导对流问题的残量型的后验误差估计．最后通过几个数值算例验证了方法的有效性．     

非定常的热传导一对流问题的特征混合元法

１．引言 设 　 Ｒ２是足够光滑的有界区域,考虑非定常的热传导－对流方程的初边值问题： 问题Ｉ．求ｕ＝（ｕ１,ｕ２）,ｐ,Ｔ满足：其中ｕ是流体的速度向量,ｐ为压力,Ｔ是温度,ｖ＞０是运动粘性系数,λ＞０是Ｇｒｏｓｈｏｆｆ数,ｊ＝（０,１）是二维向量,ｘ＝（ｘ１,ｘ２）． 非定常的热传导一对流方程是大气动力学中的一个重要的方程,这个方程组也称为强迫耗散的非线性系统方程组,其较Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程多了一个未知函数温度场,它与速度和压力之间存在着复杂的非线性关系．从热动力学可知,任何运动都会产生…     

非定常的热传导-对流问题的混合有限元-Galerkin交替方向有限元法

1 引 言设Ω　R2为具有光滑边界的有界区域,考虑非定常的,无量纲化的,而且带有热传导的粘性不可压缩流体力学问题:问题（Ⅰ）:求u=（u1,u2）,p,T满足:     

定常的Navier-Stokes方程的非线性Galerkin/Petrov最小二乘混合元法

给出定常的Navier-Stokes方程的一种非线性Galerkin/Petrov最小二乘混合元法，该方法是将余量形式的Petrov最小二乘方法与非线性Galerkin混合元结合起来，使得速度和压力的混合元空间无需满足离散的Babuska-Brezzi稳定性条件，从而使得它们的有限元空间可以任意选择。并证明该方法的解的存在唯一性和收敛性。     

热传导-对流问题的回溯二重水平法

该文给出定常的热传导-对流问题的有限元逼近的一种二重水平方法. 这种二重水平方法包括解一个小的非线性的粗网格系统、一个细网格上的线性Oseen问题和一个粗网格上的线性校正问题. 同时,给出了这种近似解的存在性和收敛性分析.     

非定常的热传导-对流问题的非线性Galerkin混合元法(Ⅲ):时间二阶精度的全离散格式

In this paper, a fully discrete format of nonlinear Galerkin mixed element method with two-step discretization of time for the non stationary conduction-convection problems is presented. The existence and the convergence of the fully discrete mixed element solution are shown. On the basis of [9] and [10], we have proved that the schemes have second-order convergence accuracy for the time discretization.     

对流占优Sobolev方程的最小二乘Galerkin有限元法

Sobolev方程在流体穿过裂缝岩石的渗透理论,土壤中湿气迁移问题,不同介质间的热传导问题等许多数学物理方面有着广泛的应用．在许多实际应用中,常常涉及到具有对流项Sobolev的方程．对于初值问题（1．1）,[1]用标准有限元方法对此类问题进行了很好的讨论和研究．混合元方法也逐渐应用到此类问题中,参看[2]．     

A dynamic meshless method for the least squares problem with some noisy subdomains

The interpolation method by radial basis functions is used widely for solving scattered data approximation. However, sometimes it makes more sense to approximate the solution by least squares fit. This is especially true when the data are contaminated with noise. A meshfree method namely, meshless dynamic weighted least squares (MDWLS) method, is presented in this paper to solve least squares problems with noise. The MDWLS method by Gaussian radial basis function is proposed to fit scattered data with some noisy areas in the problem’s domain. Existence and uniqueness of a solution is proved. This method has one parameter which can adjusts the accuracy according to the size of noises. Another advantage of the developed method is that it can be applied to problems with nonregular geometrical domains. The new approach is applied for some problems in two dimensions and the obtained results confirm the accuracy and efficiency of the proposed method. The numerical experiments illustrate that our MDWLS method has better performance than the traditional least squares method in case of noisy data.     

定常的磁流体力学方程的非线性Galerkin-Petrov最小二乘混合元法

本文提出了磁流体力学方程的一种非线性Galerkin-Petrov最小二乘混合元法,并导出了该方法解的存在性和误差估计.     

Preconditioned conjugate gradient method for generalized least squares problems

A variant of the preconditioned conjugate gradient method to solve generalized least squares problems is presented. If the problem is min (Ax − b)^TW⁻¹(Ax − b) with A ∈ R^m×n and W ∈ R^m×m symmetric and positive definite, the method needs only a preconditioner A₁ ∈ R^n×n, but not the inverse of matrix W or of any of its submatrices. Freund's comparison result for regular least squares problems is extended to generalized least squares problems. An error bound is also given.     

Preconditioning in H and applications

We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I. The natural setting for such problems is in the Hilbert space H and the variational formulation is based on the inner product in H. We show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques. These preconditioners are shown to be spectrally equivalent to the inverse of the operator. As a consequence, they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations, with the number independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems.

     

Least squares methods for Volterra equations and generalizations

In this article least squares approximations to Volterra integral equations are considered, both with exact integration and with quadrature. Optimal error estimates are derived, and it is shown that the same order of convergence is obtained in both cases with only modest requirements on the quadrature rule used in the latter. The most important practical setting for least squares is the case of convolution kernels, and these are also studied in this article. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 679–693, 1998