一种求解半线性椭圆问题的快速多重网格法  

AN EFFICIENT MULTIGRID METHOD FOR SEMILINEAR ELLIPTIC EQUATION

作　　者：谢和虎[1,2] 谢满庭 张宁 Xie Hehu;Xie Manting;Zhang Ning(LSEC, NCMIS, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China;Center for Applied Mathematics, Tianjin University, Tianjin 300072, China)

机构地区：[1]中国科学院数学与系统科学研究院,计算数学研究所,国家数学与交叉科学中心,科学与工程计算国家重点实验室,北京100190 [2]中国科学院大学,数学科学学院,北京100049 [3]天津大学应用数学中心,天津300072

出　　处：《数值计算与计算机应用》2019年第2期143-160,共18页Journal on Numerical Methods and Computer Applications

基　　金：国家自然科学基金(91730302,11771434,91330202,11371026,11001259,11031006)资助

摘　　要：本文介绍一种求解半线性问题的完全多重网格算法,该算法是基于多重校正算法与线性边值问题的多重网格迭代结合而设计的.多重校正算法将半线性问题的求解转化成线性边值问题的求解加上在一个低维空间上的半线性问题的求解.利用并行计算技术,这里所提出的多重网格算法可以明显地提高求解半线性椭圆问题的效率.更进一步,当非线性项是多项式函数的时候,本文也设计了一种高效的完全多重网格算法,并且通过分析可以知道该算法求解多项式形式的半线性椭圆问题的计算量具有渐近最优的性质.最后用数值实验验证了本文算法的有效性.A full multigrid method is proposed to solve the semilinear elliptic problem by the finite element method based on the combination of multilevel correction method and multigrid method for the linear elliptic problems. In the proposed method, solving the semilinear problem is decomposed into solutions of the linear elliptic problem by the multigrid method,and the semilinear problem which is defined in a very low dimension space. With the help of parallel computing technique, the overfull efficiency can be improved clearly. Furthermore, when the nonlinear term is a polynomial function, an efficient full multigrid method is designed such that the asymptotically computational work is absolutely optimal. One numerical example is provided to validate the efficiency of the proposed method in this paper.

关 键 词：半线性椭圆问题 有限元 完全多重网格 多水平校正 张量计算技术 渐近计算量绝对最优 

分 类 号：O241.82[理学—计算数学]