EFK方程一个新的低阶非协调混合有限元方法的高精度分析

对Extended Fisher-Kolmogorov(EFK)方程,利用EQ_1~(rot)元和零阶RaviartThomas(R-T)元建立了一个新的非协调混合元逼近格式.首先,证明了半离散格式逼近解的一个先验估计并证明了逼近解的存在唯一性.在半离散格式下,利用上述两种元的高精度分析结果以及这个先验估计,在不需要有限元解u_h属于L~∞的条件下,得到了原始变量u和中间变量v=-?u的H~1-模以及流量p=u的(L~2)~2-模意义下O(h~2)阶的超逼近性质.同时,借助插值后处理技术,证明了上述变量的具有O(h~2)阶的整体超收敛结果.其次,建立了一个新的线性化向后Euler全离散格式并证明了其逼近解的存在唯一性.另一方面,通过对相容误差和非线性项采取与传统误差分析不同的新的分裂技巧,分别导出了以往文献中尚未涉及的关于u和v在H~1-模以及p在(L~2)~2-模意义下具有O(h~2+τ)阶的超逼近性质,进一步地,借助插值后处理技术,得到了上述变量的整体超收敛结果.这里h和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.

High accuracy analysis of a new low order nonconforming mixed finite element method for the EFK equation

With the help of EQrot1 element and zero order Raviart-Thomas(R-T) elements, a new nonconforming mixed finite elements approximation scheme is proposed for the extended Fisher-Kolmogorov equation(EFK). Firstly, a priori estimate of approximation solutions is proved and the existence and uniqueness are also derived for semi-discrete scheme. Based on the high accuracy analysis of two elements and the priori estimate of the finite element solutions, the superclose properties of order O(h2) for the primitive solution u and the intermediate variable v = ??u in H1-norm and flux (p) = ?u in (L2)2-norm are obtained without the boundedness of numerical solution uh in L∞-norm, respectively. Meanwhile, the global superconvergent error estimates for above variables of order O(h2) are proved through interpolated postprocessing technique. Secondly, a new linearized backward Euler full-discrete scheme is established, the existence and uniqueness of approximation scheme are proved. On the other hand, the superclose properties of order O(h2+τ) for variable u, v in H1-norm and (p) in (L2)2-norm are obtained respectively through a new splitting technique for consistent error estimate and nonlinear term, which have never been considered in the previous literature. Furthermore, the global superconvergent estimates for above variables are deduced by interpolated postprocessing technique. Here, h, τ are parameters of the subdivision in space and time step, respectively. Finally, numerical results are provided to confirm the theoretical analysis.