共查询到20条相似文献,搜索用时 15 毫秒
1.
In this paper, we consider second order elliptic problems with rapidly oscillating coefficients. On basis of [O.A. Oleinik, A.S. Shamaev, G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992; Wen-ming He, Jun-zhi Cui, A pointwise estimate on the 1-order approximation of , IMA J. Appl. Math. 70 (2005) 241-269] we propose a new approximate method to solve these problems. Of course, we present its error estimate. 相似文献
2.
In solid state physics, the most widely used techniques to calculate the electronic levels in nanostructures are the effective masses approximation (EMA) and its extension the multiband k · p method (see [9]). They have been particularly successful in the case of heterostructures (see, e.g. [4], [9] and [11]). This paper discusses the multiscale analysis of the Schrödinger equation with rapidly oscillating coefficients. The new contributions obtained in this paper are the determination of the convergence rate for the approximate solutions, the definition of boundary layer solutions, and higher-order correctors. Consequently, a multiscale finite element method and some numerical results are presented. As one of the main results of this paper, we give a reasonable interpretation why the effective mass approximation is very accurate for calculating the band structures in semiconductor in the vicinity of Γ point, from the viewpoint of mathematics. 相似文献
3.
In this paper, on basis of [O.A. Oleinik, A.S. Shamaev, G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992], we present a local error estimate of the method of multi-scale asymptotic expansions for second order elliptic problems with rapidly oscillatory coefficients. 相似文献
4.
In this paper, we consider the initial-boundary value problem of parabolic type equation with rapidly oscillating coefficients in both time and space. A multiscale asymptotic expansion of solution for this kind of problem is presented. The full discrete finite element method for computing above problem is introduced. This method can apply to heat conduction analysis of composite materials. The main advantages of this method are that it can greatly save computer memory and CPU time, and it has good precision at the same time. Finally numerical results show that the method presented in this paper is effective and reliable. 相似文献
5.
This paper discusses the spectral properties and numerical simulation for the second order elliptic operators with rapidly
oscillating coefficients in the domains which may contain small cavities distributed periodically with period ε. A multiscale
asymptotic analysis formula for this problem is obtained by constructing properly the boundary layer. Finally, numerical results
are given, which provide a strong support for the analytical estimates 相似文献
6.
Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients 总被引:18,自引:0,他引:18
We propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients. The main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution. This is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions. As a consequence, the base functions are adapted to the local properties of the differential operator. In this paper, we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain a useful asymptotic solution structure. The issue of boundary conditions for the base functions will be discussed. Our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution, independently of the small scale in the homogenization limit. Application of our method to problems with continuous scales is also considered.
7.
V. B. Levenshtam 《Mathematical Notes》2006,79(5-6):675-680
The paper deals with averaging problems for parabolic equations. We prove that exponential dichotomy is preserved without any assumption concerning the almost-periodicity of the coefficients. 相似文献
8.
9.
A. S. Lyapin 《Mathematical Notes》2007,82(3-4):347-351
We consider a nonlinear nonautonomous hyperbolic equation with dissipation and with a small parameter multiplying the highest derivative with respect to time. This equation also involves a rapidly oscillating external force. Using a standard technique for constructing the trajectory attractor, we can prove the convergence of the attractor of a nonlinear nonautonomous hyperbolic equation with dissipation to the attractor of the corresponding parabolic equation. 相似文献
10.
Xiao-Qi Liu Li-Qun Cao Qi-Ding Zhu 《Journal of Computational and Applied Mathematics》2009,233(4):1823-921
In this paper, we consider the elastomechanical problems of a honeycomb structure of composite materials. A multiscale finite element method and the postprocessing technique with high accuracy are presented. We will derive the proofs of all theoretical results. Finally, some numerical tests validate the theoretical results of this paper. 相似文献
11.
带有振动系数的一类高阶中立型非线性受迫微分方程的振动准则 总被引:1,自引:0,他引:1
在本文中,我们获得形如[y(t) l∑j=1pj(t)y(σj(t))](n) ∫0q(t,s)f(y(t s))dσ(s)=h(t)的带有振动系数的一类高阶中立型非线性受迫微分方程的振动准则. 相似文献
12.
13.
In this paper, using asymptotic expansion method, we obtain accurate solutions for some nonlinear two point boundary value problems with rapidly oscillating coefficients. 相似文献
14.
R.K. Mohanty M.K. Jain Kochurani George 《Journal of Computational and Applied Mathematics》1996,70(2):231-243
In this paper, we develop implicit difference schemes of O(k4 + k2h2 + h4), where k > 0, h > 0 are grid sizes in time and space coordinates, respectively, for solving the system of two space dimensional second order nonlinear hyperbolic partial differential equations with variable coefficients having mixed derivatives subject to appropriate initial and boundary conditions. The proposed difference method for the scalar equation is applied for the solution of wave equation in polar coordinates to obtain three level conditionally stable ADI method of O(k4 + k2h2 + h4). Some physical nonlinear problems are provided to demonstrate the accuracy of the implementation. 相似文献
15.
Under appropriate assumptions the higher order energy decay rates for the damped wave equations with variable coefficients c(x)utt−div(A(x)∇u)+a(x)ut=0 in Rn are established. The results concern weighted (in time) and pointwise (in time) energy decay estimates. We also obtain weighted L2 estimates for spatial derivatives. 相似文献
16.
A system of linear differential equations with oscillatory decreasing coefficients is considered. The coefficients have the
formt
-α
a(t), α > 0 wherea(t) is a trigonometric polynomial with an arbitrary set of frequencies. The asymptotic behavior of the solutions of this system
ast → ∞ is studied. We construct an invertible (for sufficiently larget) change of variables that takes the original system to a system not containing oscillatory coefficients in its principal
part. The study of the asymptotic behavior of the solutions of the transformed system is a simpler problem. As an example,
the following equation is considered:
, where λ andα, 0 <α ≤ 1, are real numbers.
Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 658–666, November, 1998. 相似文献
17.
Bhupen Deka Tazuddin Ahmed 《Numerical Methods for Partial Differential Equations》2013,29(5):1522-1542
A finite element method is proposed and analyzed for hyperbolic problems with discontinuous coefficients. The main emphasize is given on the convergence of such method. Due to low global regularity of the solutions, the error analysis of the standard finite element method is difficult to adopt for such problems. For a practical finite element discretization, optimal error estimates in L∞(L2) and L∞(H1) norms are established for continuous time discretization. Further, a fully discrete scheme based on a symmetric difference approximation is considered, and optimal order convergence in L∞(H1) norm is established. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
18.
L.Q. Cao 《Journal of Mathematical Analysis and Applications》2008,343(2):1103-1118
In this paper, we study the optimal control on the boundary for parabolic equations with rapidly oscillating coefficients arising from the heat transfer problems and the optimal control on the boundary of composite materials or porous media. The multiscale asymptotic expansion of the solution for the problem in the case without any constraints is presented. We derive the proofs of all convergence results. 相似文献
19.
Implicit difference schemes of O(k4 + k2h2 + h4), where k0, h 0 are grid sizes in time and space coordinates respectively, are developed for the efficient numerical integration of the system of one space second order nonlinear hyperbolic equations with variable coefficients subject to appropriate initial and Dirichlet boundary conditions. The proposed difference method for a scalar equation is applied for the wave equation in cylindrical and spherical symmetry. The numerical examples are given to illustrate the fourth order convergence of the methods. 相似文献
20.
Yin Huicheng 《应用数学学报(英文版)》2000,16(3):299-312
For a class of quasilinear wave equations with small initial data, first we give the lower bound of lifespan of classical
solutions, then we discuss the long time asymptotic behaviour of solutions away from the blowup time.
This project is supported by the Tianyuan Foundation of China and Laburay of Mathematics for Nonlinear Problems, Fudan University. 相似文献