一种求解二阶椭圆微分方程反源问题的高次有限元法
发布时间：2019-01-11 10:55
【摘要】：本文针对二阶椭圆微分方程反源问题,讨论了由文[22]引入的极小问题的高次有限元方法,建立了基于全局测量数据的反源问题的两种二次有限元变分问题解函数的误差估计理论.数值实验表明,当扰动量可忽略时,两种二次有限元变分问题解函数的误差阶均高于线性元情形;当扰动量不可忽略时,在某种参数的适当选取下,二次有限元解函数的误差关于扰动量的下降率高于线性元情形.实验结果验证了误差估计的正确性.进一步,给出基于局部测量数据的反源问题的高次元算法.数值结果表明,当局部测量数据的延拓函数在子区域边界上满足一定的光滑性时,该算法是稳健的.
[Abstract]:In this paper, for the inverse source problem of second order elliptic differential equation, we discuss the high order finite element method for the minimal problem introduced in [22]. The error estimation theory of solutions to two quadratic finite element variational problems based on global measurement data is established. Numerical experiments show that when the disturbance is negligible, the error order of the solutions of the two quadratic finite element variational problems is higher than that of the linear element. When the disturbance can not be ignored, the error of quadratic finite element solution is higher than that of linear element when the parameter is properly selected. The experimental results verify the correctness of the error estimation. Furthermore, a high order algorithm for inverse source problem based on local measurement data is presented. The numerical results show that the algorithm is robust when the continuation function of the local measured data satisfies certain smoothness on the subregion boundary.
【学位授予单位】：湘潭大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
本文编号：2407060
[Abstract]:In this paper, for the inverse source problem of second order elliptic differential equation, we discuss the high order finite element method for the minimal problem introduced in [22]. The error estimation theory of solutions to two quadratic finite element variational problems based on global measurement data is established. Numerical experiments show that when the disturbance is negligible, the error order of the solutions of the two quadratic finite element variational problems is higher than that of the linear element. When the disturbance can not be ignored, the error of quadratic finite element solution is higher than that of linear element when the parameter is properly selected. The experimental results verify the correctness of the error estimation. Furthermore, a high order algorithm for inverse source problem based on local measurement data is presented. The numerical results show that the algorithm is robust when the continuation function of the local measured data satisfies certain smoothness on the subregion boundary.
【学位授予单位】：湘潭大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
【参考文献】
相关期刊论文 前3条
1 宋世俊;黄建国;;一个反源问题的极小化能量泛函正则化方法[J];上海交通大学学报;2011年12期
2 张曦;朱昌雄;冯广达;朱红惠;郭萍;;基于拟杆菌特异性16S rRNA基因的塘坝型饮用水污染溯源研究[J];农业环境科学学报;2011年09期
3 郑晓势,张玉海;二维椭圆型偏微分方程的反源问题[J];山东大学学报(自然科学版);2001年04期
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