奇异摄动问题基于多尺度有限元格式的一致超收敛分析  

SINGULARLY PERTURBED PROBLEM BASED ON MULTISCALE FINITE ELEMENT SCHEME AND ITS UNIFORM SUPER-CONVERGENCE ANALYSES

作　　者：孙美玲[1,2] 江山 黎野平[2] Sun Meiling;Jiang Shan;Li Yeping(Department of Mathematics,Nantong Vocational University,Nantong 226007,China;School of Science,Nantong University,Nantong 226019,China)

机构地区：[1]南通职业大学数学教研室,南通226007 [2]南通大学理学院,南通226019

出　　处：《计算数学》2023年第4期447-463,共17页Mathematica Numerica Sinica

基　　金：国家自然科学基金资助面上项目(11771224,12171258);南通市基础科学研究指令性项目(JC2021123)资助.

摘　　要：针对奇异摄动对流扩散边界层问题,应用多尺度有限元法结合自适应的分层网格提出逼近理论并进行数值模拟.多尺度有限元法仅需在粗尺度规模展开运算,通过多尺度基函数建立尺度之间的映射关系,实现从微观到宏观的数据嵌入.再结合分层网格用于粗单元离散化,能够自适应地逼近边界层.理论证明了多尺度有限元解的能量范数误差估计具有稳定性和超收敛,数值验证了其精确高效的一致超收敛结果.A multiscale finite element method combined with an adaptively graded mesh is proposed for a convection-diffusion problem with boundary layers in the singular perturbation.The multiscale computation just operates on the coarse-scale level.We present the detailed mapping behaviors among scales through the multiscale basis functions,and provide the enriched data from microscopic scales to macroscopic ones.Further more,the graded meshes are ready for a coarse-scale discretization,and they are capable of approximating the boundary layers adaptively.Mathematical theories for the energy norm error estimate of multiscale solutions are proved to be stable and super-convergent.In numerical experiments,the accurate and efficient uniform super-convergence results are validated by this novel multiscale scheme.

关 键 词：对流扩散方程 边界层 多尺度有限元法 自适应网格 超收敛 

分 类 号：O241.82[理学—计算数学]