带广义边界条件的四阶抛物型方程的混合间断时空有限元法

构造具有广义边界条件的四阶线性抛物型方程的混合间断时空有限元格式,利用混合有限元方法将高阶方程降阶,利用空间连续而时间允许间断的时空有限元方法离散方程,证明了离散解的存在唯一性,稳定性和收敛性,并给出数值算例验证了方法的有效性.     

一类四阶抛物型积分-微分方程的混合间断时空有限元法

构造四阶抛物型积分-微分方程的混合间断时空有限元格式,利用混合有限元方法将高阶方程降阶,利用空间连续而时间允许间断的时空有限元方法离散方程,证明离散解的稳定性,存在唯一性和收敛性.     

四阶线性抛物型积分-微分方程的混合间断时空有限元法

构造四阶抛物型积分-微分方程的混合间断时空有限元格式,利用混合有限元方法将高阶方程降阶,利用空间连续而时间允许间断的时空有限元方法离散方程,证明离散解的稳定性,存在唯一性和收敛性.     

Sobolev方程的混合连续时空有限元解的误差估计

通过引入辅助变量构造Sobolev方程的混合连续时空有限元离散格式,使得该格式既利用混合法将方程降阶,又将时间和空间两个变量同时用有限元方法离散,从而获得时空形式高精度数值模型.证明了Sobolev方程混合时空有限元解的存在唯一性、稳定性,并利用时间和空间投影算子推导出时空数值解的误差估计.     

Sobolev方程的时间间断Galerkin有限元方法

引入Sobolev方程的等价积分方程,构造Sobolev方程的新的时间间断Galerkin有限元格式.该格式不仅保持有限元解在时间剖分点处的间断特性,而且避免了传统时空有限元格式中跳跃项的出现,从而降低了格式理论分析和数值模拟的复杂性.证明了Sobolev方程的时间间断而空间连续的时空有限元解的稳定性、存在唯一性、L2...     

电报方程的时间间断时空有限元方法

为同时高精度逼近速度和位移,利用时间间断的时空有限元与降阶的思想,对一类电报方程的初边值问题建立一种时间间断时空有限元格式.利用有限差分方法与有限元方法相结合的技巧,证明了格式的稳定性和收敛性,得到了速度的L∞(L2)模和位移的L∞(H1)模最优误差估计.最后用数值算例验证了理论分析结果和所提算法的有效性.     

发展型方程的时间间断时空有限元方法

时空有限元方法通过统一时间和空间变量,克服了传统有限元方法对时间作差分离散引起的时间上的低精度,不但具有时、空高精度,而且在无结构网格上耗散特性好、无条件稳定,成为解决时间依赖问题的有效方法.本文利用抛物问题给出时间允许间断而空间连续的时空有限元方法的基本概念和过程,给出抛物型方程、积分-微分方程、双曲方程、Sobolev方程和其他高阶方程的算例,验证方法的精度和稳定性,并综合评价时间间断时空有限元方法目前的发展现状和应用前景.     

双曲松弛粘性Cahn-Hilliard方程的有限元算法

本文研究双曲松弛粘性Cahn-Hilliard方程的混合有限元数值算法.求解具有双曲松弛项和双阱势能的粘性Cahn-Hilliard方程时,在时间上采用一阶半隐格式进行离散,在空间上采用混合有限元方法进行离散.通过严格的理论分析证明了数值格式的无条件能量稳定性和误差估计,并利用数值实验验证了该方法的有效性.     

对流扩散方程的时空间断有限元方法

研究对流扩散方程的时空间断Galerkin有限元方法,该方法采用时,空两个变量都允许间断的基函数,更适用于移动网格,自适应算法以及并行计算.本文利用拉格朗日欧拉方法,采用F.Brezzi数值流通量,给出对流扩散方程的间断时空有限元离散格式,并证明格式的相容性,强制性,稳定性,解的存在唯一性,以及总体误差估计.     

非线性sine-Gordon方程的连续时空混合有限元方法

该文将混合有限元方法和连续时空有限元方法相结合,构造了sine-Gordon方程的连续时空混合有限元离散格式,引入独立变量p=u_t来求解,并将时间变量和空间变量都用有限元方法处理.此格式可以将方程降阶,降低有限元空间的光滑性要求,同时在时间和空间两个方向都能发挥有限元方法的优势,获得时空高精度的数值解.理论分析中严格证明了数值解的稳定性,给出了u和p的误差估计.最后通过数值算例的结果展示了格式的有效性和可行性.     

DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR A FORWARD-BACKWARD HEAT EQUATION

A space-time finite element method,discontinuous in time but continuous in space, is studied to solve the nonlinear forward-backward heat equation. A linearized technique is introduced in order to obtain the error estimates of the approximate solutions. And the numerical simulations are given.     

Space-Time adaptive algorithm for the mixed parabolic problem

In this paper we present an a-posteriori error estimator for the mixed formulation of a linear parabolic problem, used for designing an efficient adaptive algorithm. Our space-time discretization consists of lowest order Raviart-Thomas finite element over graded meshes and discontinuous Galerkin method with variable time step. Finally, several examples show that the proposed method is efficient and reliable.     

Dynamic analysis of porous materials: Numerical modeling with a space-time FEM

In the current work, we investigate the dynamic analysis of a two-phase porous material using the space-time discontinuous Galerkin method. The physical model is based on the Theory of Porous Media (TPM). The finite element approximation consists of continuous approximations in space but discontinuous ones in time. The continuity condition between the adjacent time intervals is weakly enforced by the upwind flux treatment. No artificial penalty function is involved. Moreover, the Embedded Velocity Integration technique is applied to reduce the second-order equation in time into a first order one without introducing an additional constraint. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition

Fully discrete discontinuous Galerkin methods with variable mesh- es in time are developed for the fourth order Cahn-Hilliard equation arising from phase transition in materials science. The methods are formulated and analyzed in both two and three dimensions, and are proved to give optimal order error bounds. This coupled with the flexibility of the methods demonstrates that the proposed discontinuous Galerkin methods indeed provide an efficient and viable alternative to the mixed finite element methods and nonconforming (plate) finite element methods for solving fourth order partial differential equations.

     

A characteristics-mixed finite element method for Burgers’ equation

In this paper, we propose a new mixed finite element method, called the characteristics-mixed method, for approximating the solution to Burgers’ equation. This method is based upon a space-time variational form of Burgers’ equation. The hyperbolic part of the equation is approximated along the characteristics in time and the diffusion part is approximated by a mixed finite element method of lowest order. The scheme is locally conservative since fluid is transported along the approximate characteristics on the discrete level and the test function can be piecewise constant. Our analysis show the new method approximate the scalar unknown and the vector flux optimally and simultaneously. We also show this scheme has much smaller time-truncation errors than those of standard methods. Numerical example is presented to show that the new scheme is easily implemented, shocks and boundary layers are handled with almost no oscillations. One of the contributions of the paper is to show how the optimal error estimates inL ²(Ω) are obtained which are much more difficult than in the standard finite element methods. These results seem to be new in the literature of finite element methods.     

KdV方程的特征线混合间断有限元方法

考虑KdV方程的两种特征线性混合间断有限元方法,一种方法建立在标准特征线修正方法的基础上,另一种方法是带有对流项修正的特征线修正方法.利用具有实际物理意义的特征线追踪技巧处理时间导数项和对流项,采用混合间断有限元方法处理三阶导数项,分别证明了两种方法有限元解的唯一性、稳定性和误差估计.     

A combined mixed and discontinuous Galerkin method for compressible miscible displacement problem in porous media

In this paper, we present a numerical scheme for solving the coupled system of compressible miscible displacement problem in porous media. The flow equation is solved by the mixed finite element method, and the transport equation is approximated by a discontinuous Galerkin method. The scheme is continuous in time and a priori hp error estimates is presented.