High order compact schemes for gradient approximation  

作　　者：Huang YunQing Liang Qin Yi NianYu 

机构地区：[1]Xiangtan Univ, Sch Math & Computat Sci, Hunan Key Lab Computat & Simulat Sci & Engn, Xiangtan 411105, Peoples R China [2]Peking Univ, Sch Math Sci, Beijing 100871, Peoples R China

出　　处：《Science China Mathematics》2010年第7期1899-1914,共16页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No. 10625106);the National Basic Research Program of China (Grant No. 2005CB321701);supported by Graduate School Visit Project of Peking University;Hunan Provincial Innovation Foundation for Postgraduate (Grant No. S2008yjscx05)

摘　　要：In this paper, we propose three gradient recovery schemes of higher order for the linear interpolation. The first one is a weighted averaging method based on the gradients of the linear interpolation on the uniform mesh, the second is a geometric averaging method constructed from the gradients of two cubic interpolation on macro element, and the last one is a local least square method on the nodal patch with cubic polynomials. We prove that these schemes can approximate the gradient of the exact solution on the symmetry points with fourth order. In particular, for the uniform mesh, we show that these three schemes are the same on the considered points. The last scheme is more robust in general meshes. Consequently, we obtain the superconvergence results of the recovered gradient by using the aforementioned results and the supercloseness between the finite element solution and the linear interpolation of the exact solution. Finally, we provide several numerical experiments to illustrate the theoretical results.In this paper, we propose three gradient recovery schemes of higher order for the linear interpolation. The first one is a weighted averaging method based on the gradients of the linear interpolation on the uniform mesh, the second is a geometric averaging method constructed from the gradients of two cubic interpolation on macro element, and the last one is a local least square method on the nodal patch with cubic polynomials. We prove that these schemes can approximate the gradient of the exact solution on the symmetry points with fourth order. In particular, for the uniform mesh, we show that these three schemes are the same on the considered points. The last scheme is more robust in general meshes. Consequently, we obtain the superconvergence results of the recovered gradient by using the aforementioned results and the supercloseness between the finite element solution and the linear interpolation of the exact solution. Finally, we provide several numerical experiments to illustrate the theoretical results.

关 键 词：SUPERCONVERGENCE GRADIENT RECOVERY COMPACT SCHEME 

分 类 号：N[自然科学总论]