A full multigrid method for nonlinear eigenvalue problems  

作　　者：JIA ShangHui XIE HeHu XIE ManTing XU Fei 

机构地区：[1]School of Statistics and Mathematics, Central University of Finance and Economics [2]LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences

出　　处：《Science China Mathematics》2016年第10期2037-2048,共12页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant Nos. 91330202, 11371026, 11201501, 11571389, 11001259 and 11031006);National Basic Research Program of China (Grant No. 2011CB309703);the National Center for Mathematics and Interdisciplinary Science, Chinese Academy of Sciences, the President Foundation of Academy of Mathematics and Systems Science, Chinese Academy of Sciences and the Program for Innovation Research in Central University of Finance and Economics

摘　　要：We introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of the nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations.Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We prove that the computational work of this new scheme is truly optimal,the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.We introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of the nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We prove that the computational work of this new scheme is truly optimal, the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.

关 键 词：nonlinear eigenvalue problem full multigrid method multilevel correction finite element method 

分 类 号：O241.82[理学—计算数学]