多尺度有限元结合Bakhvalov-Shishkin网格法高效处理边界层问题的研究  

Combining the multiscale finite element and Bakhvalov-Shishkin grid to solve the boundary layer problems

作　　者：江山[1] 孙美玲[1,2] 

机构地区：[1]扬州大学数学科学学院,江苏扬州225002 [2]南通职业大学,江苏南通226007

出　　处：《浙江大学学报（理学版）》2015年第2期142-146,共5页Journal of Zhejiang University（Science Edition）

基　　金：国家自然科学青年基金资助项目(11301462);江苏省高校自然科学研究面上项目(13KJB110030);江苏省高校研究生科研创新计划项目(KYLX_1332);扬州大学博士后研究项目;扬州大学新世纪人才工程资助项目

摘　　要：提出了用多尺度有限元逼近法来模拟奇异摄动的对流扩散边界层问题.通过求解基于微分算子的子问题获得的多尺度基函数来有效捕获边界层的局部信息,用改良的Bakhvalov-Shishkin(B-S)网格来求解奇异摄动的对流扩散边界层问题,可实现高效逼近.与传统有限元法相比,多尺度有限元法占用的计算资源和存储空间较少,利用B-S粗网格就可得到不依赖于小参数ε、精度很高的2阶L2范数的一致超收敛结果.特别当参数ε非常小时,采用多尺度有限元结合B-S网格来求解奇异摄动问题,优势更显著.An efficient multiscale finite element method is proposed to simulate the singularly perturbed convection-diffusion model with the boundary layers.We capture the boundary local information effectively by solving the multiscale basis functions obtained from differential operator subproblems,and construct the modified Bakhvalov-Shishkin（B-S）grid on the coarse meshes as well.Compared with the traditional finite element methods,our new method only costs quite a few computer resources and memory space.The application of the B-S grid can acquire the superconvergence results independent of the parameterεwith the accurate second normL2.The numerical experiment demonstrates the application advantages of the multiscale computation which is combined with the B-S grid to deal with the singular perturbation problems,especially for a very small parameterε.

关 键 词：多尺度有限元 Bakhvalov-Shishkin网格 奇异摄动 边界层 对流扩散 

分 类 号：O241.82[理学—计算数学]