A two-grid partition of unity method for second order elliptic problems is proposed and analyzed. The standard two-grid method is a local and parallel method usually leading to a discontinuous solution in the entire c...A two-grid partition of unity method for second order elliptic problems is proposed and analyzed. The standard two-grid method is a local and parallel method usually leading to a discontinuous solution in the entire computational domain. Partition of unity method is employed to glue all the local solutions together to get the global continuous one, which is optimal in HI-norm. Furthermore, it is shown that the L^2 error can be improved by using the coarse grid correction. Numerical experiments are reported to support the theoretical results.展开更多
We obtain a priori estimates and solvability in Hardy type space in a bounded domain of Rn for second order elliptic equations with coefficients of limited smoothness. Such a result can be served as an endpoint case o...We obtain a priori estimates and solvability in Hardy type space in a bounded domain of Rn for second order elliptic equations with coefficients of limited smoothness. Such a result can be served as an endpoint case of the classical LP（1 〈 p 〈 ∞） theory for second order elliptic equations. Our approach is based on a standard technique of perturbation rather than that of integral representation formula.展开更多
In this paper, the method of non-conforming mixed finite element for second order elliptic problems is discussed and a format of real optimal order for the lowest order error estimate.
The existence of positive radial solutions to the second order semilinear elliptic BVPΔu(X)+g(|X|)f(u(X))=0, R 1<|X|<R 2, u(X)=0, |X|=R 1 or |X|=R 2is considered. A general existence criterion and se...The existence of positive radial solutions to the second order semilinear elliptic BVPΔu(X)+g(|X|)f(u(X))=0, R 1<|X|<R 2, u(X)=0, |X|=R 1 or |X|=R 2is considered. A general existence criterion and several existence theorems of positive radial solution are established. Here it is not required that lim l→0f(l)/l and lim l→∞f(l)/l exist.展开更多
This paper is devoted to study of an iterative procedure for domain decomposition method of second order elliptic problem with mixed boundary conditions (i.e., Dirichlet condition on a part of boundary and Neumann con...This paper is devoted to study of an iterative procedure for domain decomposition method of second order elliptic problem with mixed boundary conditions (i.e., Dirichlet condition on a part of boundary and Neumann condition on the another part of boundary). For the pure Dirichlet problem, Marini and Quarteroni [3], [4] considered a similar approach, which is extended to more complex problem in this paper.展开更多
We present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of （D（h） order...We present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of （D（h） order in energy norm and of O（h2） order in L2 norm on general d-rectangular triangulations. Moreover, when the triangulation is uniform, the convergence rate can be of O（h2） order in energy norm, and the convergence rate in L2 norm is still of O（h2） order, which cannot be improved. Numerical examples are presented to demonstrate our theoretical results.展开更多
In this paper,a new hybridized mixed formulation of weak Galerkin method is studied for a second order elliptic problem.This method is designed by approximate some operators with discontinuous piecewise polynomials in...In this paper,a new hybridized mixed formulation of weak Galerkin method is studied for a second order elliptic problem.This method is designed by approximate some operators with discontinuous piecewise polynomials in a shape regular finite element partition.Some discrete inequalities are presented on discontinuous spaces and optimal order error estimations are established.Some numerical results are reported to show super convergence and confirm the theory of the mixed weak Galerkin method.展开更多
Some oscillation theorems are given for the nonlinear second order elliptic equation N∑i,j=1 Di[aij（x）ψ（y）｜｜△↓y｜｜^p-2 Djy]＋c（x）f（y）=0The results are extensions of modified Riccati techniques and inclu...Some oscillation theorems are given for the nonlinear second order elliptic equation N∑i,j=1 Di[aij（x）ψ（y）｜｜△↓y｜｜^p-2 Djy]＋c（x）f（y）=0The results are extensions of modified Riccati techniques and include recent results of Usami.展开更多
In this paper,a residual type of a posteriori error estimator for the general second order elliptic eigenpair approximation by the mixed finite element method is derived and analyzed,based on a type of superconvergenc...In this paper,a residual type of a posteriori error estimator for the general second order elliptic eigenpair approximation by the mixed finite element method is derived and analyzed,based on a type of superconvergence result of the eigenfunction approximation.Its efficiency and reliability are proved by both theoretical analysis and numerical experiments.展开更多
基金Project supported by the National Natural Science Foundation of China(No.40074031)the Science Foundation of the Science and Technology Commission of Shanghai Municipalitythe Program for Young Excellent Talents in Tongji University(No.2007kj008)
文摘A two-grid partition of unity method for second order elliptic problems is proposed and analyzed. The standard two-grid method is a local and parallel method usually leading to a discontinuous solution in the entire computational domain. Partition of unity method is employed to glue all the local solutions together to get the global continuous one, which is optimal in HI-norm. Furthermore, it is shown that the L^2 error can be improved by using the coarse grid correction. Numerical experiments are reported to support the theoretical results.
基金Supported by NNSF of China Grant No.10571084NNSF of China Grant No.10771097
文摘We obtain a priori estimates and solvability in Hardy type space in a bounded domain of Rn for second order elliptic equations with coefficients of limited smoothness. Such a result can be served as an endpoint case of the classical LP（1 〈 p 〈 ∞） theory for second order elliptic equations. Our approach is based on a standard technique of perturbation rather than that of integral representation formula.
基金Project supported by the Cultivating Foundation of Youthful Backbone of Science and Technologyof Beijing, the National Science
文摘In this paper, the method of non-conforming mixed finite element for second order elliptic problems is discussed and a format of real optimal order for the lowest order error estimate.
文摘The existence of positive radial solutions to the second order semilinear elliptic BVPΔu(X)+g(|X|)f(u(X))=0, R 1<|X|<R 2, u(X)=0, |X|=R 1 or |X|=R 2is considered. A general existence criterion and several existence theorems of positive radial solution are established. Here it is not required that lim l→0f(l)/l and lim l→∞f(l)/l exist.
文摘This paper is devoted to study of an iterative procedure for domain decomposition method of second order elliptic problem with mixed boundary conditions (i.e., Dirichlet condition on a part of boundary and Neumann condition on the another part of boundary). For the pure Dirichlet problem, Marini and Quarteroni [3], [4] considered a similar approach, which is extended to more complex problem in this paper.
基金supported by National Natural Science Foundation of China (Grant Nos. 11471026, 11271035, 91430213, 11421101 and 11101415)
文摘We present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of （D（h） order in energy norm and of O（h2） order in L2 norm on general d-rectangular triangulations. Moreover, when the triangulation is uniform, the convergence rate can be of O（h2） order in energy norm, and the convergence rate in L2 norm is still of O（h2） order, which cannot be improved. Numerical examples are presented to demonstrate our theoretical results.
文摘In this paper,a new hybridized mixed formulation of weak Galerkin method is studied for a second order elliptic problem.This method is designed by approximate some operators with discontinuous piecewise polynomials in a shape regular finite element partition.Some discrete inequalities are presented on discontinuous spaces and optimal order error estimations are established.Some numerical results are reported to show super convergence and confirm the theory of the mixed weak Galerkin method.
文摘Some oscillation theorems are given for the nonlinear second order elliptic equation N∑i,j=1 Di[aij（x）ψ（y）｜｜△↓y｜｜^p-2 Djy]＋c（x）f（y）=0The results are extensions of modified Riccati techniques and include recent results of Usami.
基金supported by National Natural Science Foundation of China(Grant Nos.11001259,11031006,11071265,11201501 and 91230110)National Basic Research Program of China(973 Project)(Grant No. 2011CB309703)+3 种基金International S&T Cooperation Program of China(Grant No. 2010DFR00700)Croucher Foundation of Hong Kong Baptist Universitythe National Center for Mathematics and Interdisciplinary Science,CAS,the President Foundation of AMSS-CASthe Fundamental Research Funds for the CentralUniversities(Grant No. 2012121003)
文摘In this paper,a residual type of a posteriori error estimator for the general second order elliptic eigenpair approximation by the mixed finite element method is derived and analyzed,based on a type of superconvergence result of the eigenfunction approximation.Its efficiency and reliability are proved by both theoretical analysis and numerical experiments.
