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 ...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.展开更多
A cascadic multigrid method is proposed for eigenvalue problems based on the multilevel correction scheme. With this new scheme, an eigenvalue problem on the finest space can be solved by linear smoothing steps on a s...A cascadic multigrid method is proposed for eigenvalue problems based on the multilevel correction scheme. With this new scheme, an eigenvalue problem on the finest space can be solved by linear smoothing steps on a series of multilevel finite element spaces and nonlinear correcting steps on special coarsest spaces. Once the sequence of finite element spaces and the number of smoothing steps are appropriately chosen, the optimal convergence rate with the optimal computational work can be obtained. Some numerical experiments are presented to validate our theoretical analysis.展开更多
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
文摘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.
基金Acknowledgments. This work is supported in part by the National Natural Science Foundation of China （NSFC 91330202, 11371026, 11001259, 11031006, 2011CB309703） and the National Center for Mathematics and Interdisciplinary Science, CAS.
文摘A cascadic multigrid method is proposed for eigenvalue problems based on the multilevel correction scheme. With this new scheme, an eigenvalue problem on the finest space can be solved by linear smoothing steps on a series of multilevel finite element spaces and nonlinear correcting steps on special coarsest spaces. Once the sequence of finite element spaces and the number of smoothing steps are appropriately chosen, the optimal convergence rate with the optimal computational work can be obtained. Some numerical experiments are presented to validate our theoretical analysis.
