Extended Fisher-Kolmogorov方程的一类低阶非协调混合有限元方法

A Type of New Lower Order Nonconforming Mixed Finite Elements Methods for the Extended Fisher-Kolmogorov Equation

该文的主要目的是研究Extended Fisher-Kolmogorov(EFK)方程的一类低阶非协调元混合有限元方法.首先引入一个中间变量v=-△u将原方程分裂为两个二阶方程,建立了一个非协调混合元逼近格式,并通过构造一个李雅普诺夫泛函证明了半离散格式逼近解的一个先验估计并证明了解的存在唯一性.在半离散格式下,利用这个先验估计和单元的性质,证明了原始变量u和中间变量v的H^1-模意义下的最优误差估计.进一步地,借助高精度技巧得到了O(h^2)阶的超逼近性质.其次,建立了一个新的线性化的向后Euler全离散格式,通过对相容误差和非线性项采用新的分裂技术,导出了u和v的H^1-模意义下具有O(h+τ)和O(h^2+τ)的最优误差估计和超逼近结果.这里,h,τ分别表示空间剖分参数和时间步长.最后,给出了一个数值算例,计算结果验证了理论分析的正确性,该文的分析为利用非协调混合有限元研究其它四阶初边值问题提供了一个可借鉴的途径.

The purpose of this paper is to study the mixed finite element methods with a type of lower order nonconforming finite elements for the extended Fisher-Kolmogorov（EFK）equation. Firstly, a nonconforming mixed finite element scheme is established by splitting the EFK equation into two second order equations through a intermediate variable v =-△u. Some a priori bounds are derived by use of Lyapunov functional, existence and uniqueness for the approximation solutions are also proved. The optimal error estimates for both the primitive solution u and the intermediate variable v in H^1-norm are deduced for semi-discrete scheme by use of the above priori bounds and properties of the elements. Furthermore, the superclose properties with order O（h^2） are obtained through high accuracy results of the elements. Secondly, a new linearized backward Euler full-discrete scheme is established. The optimal error estimates and superclose results for u and v in H^1-norm with orders O（h＋τ） and O（h^2 ＋τ） are obtained respectively through the new splitting techniques for consistency errors and nonlinear terms. Here, h, τ are parameter of the subdivision in space and time step. Finally, numerical results are provided to confirm the theoretical analysis. Our analysis provides a new understanding with nonconforming mixed finite element methods to analyze other fourth order initial boundary value problems.