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 ...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.展开更多
Instead of most existing postprocessing schemes, a new preprocessing approach, called multi- neighboring grids （MNG）, is proposed for solving PDE eigen-problems on an existing grid G（A）. The linear or multi-linear...Instead of most existing postprocessing schemes, a new preprocessing approach, called multi- neighboring grids （MNG）, is proposed for solving PDE eigen-problems on an existing grid G（A）. The linear or multi-linear element, based on box-splines, are taken as the first stage Khuh -λh/1Mh/1Uh. In this paper, the j-th stage neighboring-grid scheme is defined as Khuh λh/j Mh/j Uh = λh/j Mh/j Uh , where gh ：- Mh/j-1 Kh/1 and Mhuh is to be found as a better mass distribution over the j-th stage neighboring-grid G（/k）, and Kh/1 can be seen as an expansion of Kh on the j-th neighboring-grid with respect to the （j - 1）-th mass distribution Mh_l. It is shown that for an ODE model eigen-problem, the j-th stage scheme with 2j-th order B-spline basis can reach 2j-th order accuracy and even （2j ＋ 2）-th order accuracy by perturbing the mass matrix. The argument can be extended to high dimensions with separable variable cases. For Laplace eigen-problems with some 2-D and 3-D structured uniform grids, some 2j-th order schemes are presented for j ≤ 3.展开更多
In this paper we provide a sufficient and necessary condition for the eigenvalueand eigenvector of a general reducible matrix in a discrete-event system described by the“max”algebra,analyse the steady periodical per...In this paper we provide a sufficient and necessary condition for the eigenvalueand eigenvector of a general reducible matrix in a discrete-event system described by the“max”algebra,analyse the steady periodical performance of the system,and obtain ananalytic solution of the dynamic equation.We propose the conception of“order-d-(?)-block-periodical matrix”and obtain its sufficient and necessary condition and provide an algorithmof (?) matrix.展开更多
1 Problem and algorithmMANY computing tasks arising from computational physics, chemistry and biology are relatedto solving so-called generalized eigen-decomposition problems. Consider the following general-ized eigen...1 Problem and algorithmMANY computing tasks arising from computational physics, chemistry and biology are relatedto solving so-called generalized eigen-decomposition problems. Consider the following general-ized eigen-decomposition problem:展开更多
A new class of three-variable orthogonal polynomials, defined as eigenfunctions of a second order PDE operator, is studied. These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a map...A new class of three-variable orthogonal polynomials, defined as eigenfunctions of a second order PDE operator, is studied. These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron, and can be taken as an extension of the 2-D Steiner domain. The polynomials can be viewed as Jacobi polynomials on such a domain. Three-term relations are derived explicitly. The number of the individual terms, involved in the recurrences relations, are shown to be independent on the total degree of the polynomials. The numbers now are determined to be five and seven, with respect to two conjugate variables z, $ \bar z $ and a real variable r, respectively. Three examples are discussed in details, which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds, and Legendre polynomials.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos.60970089,61170075 and 91230109)
文摘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.
基金supported by National Natural Science Foundation of China(Grant Nos.6097008961170075 and 91230109)
文摘Instead of most existing postprocessing schemes, a new preprocessing approach, called multi- neighboring grids （MNG）, is proposed for solving PDE eigen-problems on an existing grid G（A）. The linear or multi-linear element, based on box-splines, are taken as the first stage Khuh -λh/1Mh/1Uh. In this paper, the j-th stage neighboring-grid scheme is defined as Khuh λh/j Mh/j Uh = λh/j Mh/j Uh , where gh ：- Mh/j-1 Kh/1 and Mhuh is to be found as a better mass distribution over the j-th stage neighboring-grid G（/k）, and Kh/1 can be seen as an expansion of Kh on the j-th neighboring-grid with respect to the （j - 1）-th mass distribution Mh_l. It is shown that for an ODE model eigen-problem, the j-th stage scheme with 2j-th order B-spline basis can reach 2j-th order accuracy and even （2j ＋ 2）-th order accuracy by perturbing the mass matrix. The argument can be extended to high dimensions with separable variable cases. For Laplace eigen-problems with some 2-D and 3-D structured uniform grids, some 2j-th order schemes are presented for j ≤ 3.
文摘In this paper we provide a sufficient and necessary condition for the eigenvalueand eigenvector of a general reducible matrix in a discrete-event system described by the“max”algebra,analyse the steady periodical performance of the system,and obtain ananalytic solution of the dynamic equation.We propose the conception of“order-d-(?)-block-periodical matrix”and obtain its sufficient and necessary condition and provide an algorithmof (?) matrix.
文摘1 Problem and algorithmMANY computing tasks arising from computational physics, chemistry and biology are relatedto solving so-called generalized eigen-decomposition problems. Consider the following general-ized eigen-decomposition problem:
基金the Major Basic Project of China(Grant No.2005CB321702)the National Natural Science Foundation of China(Grant Nos.10431050,60573023)
文摘A new class of three-variable orthogonal polynomials, defined as eigenfunctions of a second order PDE operator, is studied. These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron, and can be taken as an extension of the 2-D Steiner domain. The polynomials can be viewed as Jacobi polynomials on such a domain. Three-term relations are derived explicitly. The number of the individual terms, involved in the recurrences relations, are shown to be independent on the total degree of the polynomials. The numbers now are determined to be five and seven, with respect to two conjugate variables z, $ \bar z $ and a real variable r, respectively. Three examples are discussed in details, which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds, and Legendre polynomials.
基金Supported in part by the Natural Science Foundation of China(11061008)Guangxi Provincial Natural Science Foundation of China(2011GXNSFA018128)Guangxi Provincial Key Scientific Research Project of China(1355010-8)
