Extended Fisher-Kolmogorov方程的间断有限元分析

摘要/Abstract

摘要：

利用Wilson元研究了Extended Fisher-Kolmogorov（EFK）方程的间断有限元逼近格式.在不需要后处理技术的前提下，通过对非线性项采用新的分裂技术，分别得到了半离散和线性化欧拉全离散格式下原始变量 $u$ 和中间变量 $v=-\triangle u$ 的 $O(h^{2})$ 和 $O(h^{2}+\tau)$ 阶的收敛性结果，正好比通常的关于Wilson元的误差估计高出一阶.这里， $h$ ， $\tau$ 表示空间剖分参数和时间步长.

关键词: EFK方程, 间断有限元, Wilson元, 半离散和全离散格式, 收敛性

Abstract:

The discontinuous Galerkin finite element approximation schemes for the Extended Fisher-Kolmogorov (EFK) equation are studied by using the Wilson element. Without using the technique of postprocessing technique, the convergence results with order $O(h^{2})/O(h^{2}+\tau)$ for the primitive solution $u$ and intermediate variable $v=-\triangle u$ are obtained for the semi-discrete and linearized Euler fully discrete approximation schemes respectively through a new splitting technique for the nonlinear terms. The above results are just one order higher than the usual error estimates of the Wilson element. Here, $h$ and $\tau$ are parameters of the subdivision in space and time step, respectively.

Key words: EFK equation, Discontinuous Galerkin finite element, Wilson element, Semi-discrete and fully-discrete schemes, Convergence

中图分类号:

O242.21

杨晓侠,张厚超. Extended Fisher-Kolmogorov方程的间断有限元分析[J]. 数学物理学报, 2021, 41(6): 1880-1896.

Xiaoxia Yang,Houchao Zhang. Discontinuous Galerkin Finite Element Analysis of for the Extended Fisher-Kolmogorov Equation[J]. Acta mathematica scientia,Series A, 2021, 41(6): 1880-1896.

图/表 5

表 1

$t = 1$ 时的数值计算结果( $\tau = h^2)$ "

$m^{2}$	$\left\\| {u^{n}-U_h^{n} } \right\\|_h$	收敛阶	$\left\\| {v^{n}-V_h^{n} } \right\\|_h$	收敛阶
$4^{2}$	0.000046035	—	0.005970921	—
$8^{2}$	0.000012875	1.8381	0.001790764	1.7374
$16^{2}$	0.000003264	1.9799	0.000472460	1.9223
$32^{2}$	0.000000793	2.0417	0.000117908	2.0025
$64^{2}$	0.000000195	2.0240	0.000029285	2.0094

表 1

图 1

$u$ 的图( $t = 1, m^{2} = 32^{2}$ )"

图 1

图 2

数值解 $U_h$ 的图( $t = 1, m^{2} = 32^{2}$ )"

图 2

图 3

$v$ 的图( $t = 1, m^{2} = 32^{2}$ )"

图 3

图 4

数值解 $V_h$ 的图( $t = 1, m^{2} = 32^{2}$ )"

图 4

参考文献 33

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