New schemes with fractal error compensation for PDE eigenvalue computations  

作　　者：SUN JiaChang 

机构地区：[1]Laboratory of Computational Sciences,Institute of Software,Chinese Academy of Sciences

出　　处：《Science China Mathematics》2014年第2期221-244,共24页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant Nos.60970089,61170075 and 91230109)

摘　　要：With an error compensation term in the fractal Rayleigh quotient of PDE eigen-problems,we propose a new scheme by perturbing the mass matrix Mhto Mh=Mh+Ch2mKh,where Khis the corresponding stif matrix of a 2m 1 degree conforming finite element with mesh size h for a 2m-order self-adjoint PDE,and the constant C exists in the priority error estimationλh jλj^Ch2mλ2j.In particular,for Laplace eigenproblems over regular domains in uniform mesh,e.g.,cube,equilateral triangle and regular hexagon,etc.,we find the constant C=I h 1Mh2 hKh and show that in this case the computation accuracy can raise two orders,i.e.,fromλh jλj=O(h2)to O(h4).Some numerical tests in 2-D and 3-D are given to verify the above arguments.With an error compensation term in the fractal Rayleigh quotient of PDE eigen-problems,we propose a new scheme by perturbing the mass matrix Mhto Mh=Mh＋Ch2mKh,where Khis the corresponding stif matrix of a 2m 1 degree conforming finite element with mesh size h for a 2m-order self-adjoint PDE,and the constant C exists in the priority error estimationλh jλj^Ch2mλ2j.In particular,for Laplace eigenproblems over regular domains in uniform mesh,e.g.,cube,equilateral triangle and regular hexagon,etc.,we find the constant C=I h 1Mh2 hKh and show that in this case the computation accuracy can raise two orders,i.e.,fromλh jλj=O（h2）to O（h4）.Some numerical tests in 2-D and 3-D are given to verify the above arguments.

关 键 词：PDE eigenvalues computation generalized matrix eigen-problem discrete Rayleigh quotient 

分 类 号：O175.2[理学—数学] O241.6[理学—基础数学]