四阶抛物偏微分方程的H^1-Galerkin混合元方法及数值模拟

H^1-GALERKIN MIXED ELEMENT METHOD AND NUMERICAL SIMULATION FOR THE FOURTH-ORDER PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

到目前为止,H^1-Galerkin混合有限元方法研究的问题仅局限于二阶发展方程.然而对于高阶发展方程,特别是重要的四阶发展方程问题的研究却没有出现.本文首次提出四阶发展方程的H^1-Galerkin混合有限元方法,为了给出理论分析的需要,我们考虑四阶抛物型发展方程.通过引进三个适当的中间辅助变量,形成四个一阶方程组成的方程组系统,提出四阶抛物型方程的H^1-Galerkin混合有限元方法.得到了一维情形下的半离散和全离散格式的最优收敛阶误差估计和多维情形的半离散格式误差估计,并采用迭代方法证明了全离散格式的稳定性.最后,通过数值例子验证了提出算法的可行性.在一维情况下我们能够同时得到未知纯量函数、一阶导数、负二阶导数和负三阶导数的最优逼近解,这一点是以往混合元方法所不能得到的.

So far, the H^1-Galerkin mixed finite element method was applied to many second- order evolution equations. However, the H^1-Galerkin mixed method for the higher-order evolution equations, especially, for fourth-order evolution equations have not been studied in the literature. In this paper, we first proposed the H^1-Galerkin mixed method for fourth- order evolution equation. For the need of the analysis of theories, we consider the fourth- order parabolic evolution equation. By introducing three auxiliary variables, the first-order system of four equations is formulated, and the H^l-Galerkin mixed finite element method for fourth-order parabolic equation is proposed. Optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension, and error estimates are derived for semidiscrete scheme for several space dimensions, and the stability for fully discrete scheme is proved by the iteration method. Finally, some numerical results are provided to illustrate the effectiveness of our method. Optimal approximate solutions for the scalar unknown, first derivative, negative second derivative and negative third derivative are obtained, which can＇t be derived by the other mixed methods.