In this paper we study the higher accuracy methods - the extrapolation and defect correction for the semidiscrete Galerkin approximations to the solutions of the parabolic equations with boundary integral conditions. ...In this paper we study the higher accuracy methods - the extrapolation and defect correction for the semidiscrete Galerkin approximations to the solutions of the parabolic equations with boundary integral conditions. The global extrapolation and the correction approximations, rather than the pointwise extrapolation results, are derived.展开更多
Asymptotic error expansions in H^1-norm for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectan- gular meshes, the Richardson extrapolat...Asymptotic error expansions in H^1-norm for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectan- gular meshes, the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied. The higher order numerical approximations are used to generate a posteriori error estimators for the finite element approximation.展开更多
This paper develops a new method to analyze convergence of the iterated defect correction scheme of finite element methods on rectangular grids in both two and three dimensions. The main idea is to formulate energy in...This paper develops a new method to analyze convergence of the iterated defect correction scheme of finite element methods on rectangular grids in both two and three dimensions. The main idea is to formulate energy inner products and energy （semi）norms into matrix forms. Then, two constants of two key inequalities involved are min and max eigenvalues of two associated generalized eigenvalue problems, respectively. Local versions on the element level of these two generalized eigenvalue problems are exactly solved to obtain sharp （lower） upper bounds of these two constants. This and some essential observations for iterated solutions establish convergence in 2D and the monotone decreasing property in 3D. For two dimensions the results herein improve those in literature; for three dimensions the results herein are new. Numerical results are presented to examine theoretical results.展开更多
Objective The purpose of this study is to investgate changes of cTnI in myocardial ischemic and reperfusion injury during correction of cardiac defects in children. Methods From June, 1999 to May,2000,45 children (30 ...Objective The purpose of this study is to investgate changes of cTnI in myocardial ischemic and reperfusion injury during correction of cardiac defects in children. Methods From June, 1999 to May,2000,45 children (30 male, 15 female) undergoing correction of cardiac defects were divided into three groups randomly: group Ⅰ no myocardial ischemia,group Ⅱ myocardial ischemia less than 60 minutes, group Ⅲmyocardial ischemia 】 60 minutes. There were no significant differences in the three groups in age, sex ratio, C/T ratio, or left ventricular function. Blood samples for analysis were collected before skin incision and at time intervals up to 6 days postoperatively. Analysis of creatine kinase MB.LDH and cardiac-specific troponin I was used for the detection of myocardial damage. Meantime, the ECG was checked for myocardial infarction. After the reperfusion, myocardial tissue was obtained from the free wall of right ventricle myocardial structure studies. Results The level of cTnI was increased展开更多
In this paper we employ the Petrov Galerkin method for the parabolic problems to get the finite element approximate solution of high accuracy by means of the interpolation postprocessing, extrapolation and defect cor...In this paper we employ the Petrov Galerkin method for the parabolic problems to get the finite element approximate solution of high accuracy by means of the interpolation postprocessing, extrapolation and defect correction techniques.展开更多
In this paper, we study an efficient asymptotically correction of a-posteriori er- ror estimator for the numerical approximation of Volterra integro-differential equations by piecewise polynomial collocation method. T...In this paper, we study an efficient asymptotically correction of a-posteriori er- ror estimator for the numerical approximation of Volterra integro-differential equations by piecewise polynomial collocation method. The deviation of the error for Volterra integro- differential equations by using the defect correction principle is defined. Also, it is shown that for m degree piecewise polynomial collocation method, our method provides O（hm＋l） as the order of the deviation of the error. The theoretical behavior is tested on examples and it is shown that the numerical results confirm the theoretical part.展开更多
We present a class of arbitrarily high order fully explicit kinetic numerical methods in compressible fluid dynamics,both in time and space,which include the relaxation schemes by Jin and Xin.These methods can use the...We present a class of arbitrarily high order fully explicit kinetic numerical methods in compressible fluid dynamics,both in time and space,which include the relaxation schemes by Jin and Xin.These methods can use the CFL number larger or equal to unity on regular Cartesian meshes for the multi-dimensional case.These kinetic models depend on a small parameter that can be seen as a"Knudsen"number.The method is asymptotic preserving in this Knudsen number.Also,the computational costs of the method are of the same order of a fully explicit scheme.This work is the extension of Abgrall et al.(2022)[3]to multidimensional systems.We have assessed our method on several problems for two-dimensional scalar problems and Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.展开更多
In this paper we introduce a Petrov-Galerkin approximation model to the solution of linear and semi-linear elliptic boundary value problems in which piecewise quadratic polynomial space and piecewise linear polynomial...In this paper we introduce a Petrov-Galerkin approximation model to the solution of linear and semi-linear elliptic boundary value problems in which piecewise quadratic polynomial space and piecewise linear polynomial space are used as the shape function space and the test function space, respectively. We prove that the approximation order of the standard quadratic finite element can be attained in this Petrov-Galerkin model. Based on the so-called 'contractivity' of the interpolation operator, we further prove that the defect iterative sequence of the linear finite element solution converge to the proposed Petrov-Galerkin approximate solution.展开更多
In this paper, the Wilson nonconforming finite element is considered for solving a class of second-order elliptic boundary value problems. Based on an asymptotic error expansion for the Wilson finite element, the glob...In this paper, the Wilson nonconforming finite element is considered for solving a class of second-order elliptic boundary value problems. Based on an asymptotic error expansion for the Wilson finite element, the global superconvergences, the local superconvergences and the defect correction schemes are presented.展开更多
Focuses on a study which determined the use of the global convergences of the domain decomposition methods with Lagrangian multiplier and nonmatching grids in solving the second order elliptic boundary value problems....Focuses on a study which determined the use of the global convergences of the domain decomposition methods with Lagrangian multiplier and nonmatching grids in solving the second order elliptic boundary value problems. Background on domain decomposition and global superconvergence; Correction scheme and estimates; Numerical examples.展开更多
In this paper we study the higher accuracy methods-the extrapolation and defect correction for the semidiscrete Galerkin approximations to the solutions of the parabolic equations. Theglobal extrapolation and the corr...In this paper we study the higher accuracy methods-the extrapolation and defect correction for the semidiscrete Galerkin approximations to the solutions of the parabolic equations. Theglobal extrapolation and the correction approximations of third order, rather than the pointwiseextrapolation results, are derived.展开更多
文摘In this paper we study the higher accuracy methods - the extrapolation and defect correction for the semidiscrete Galerkin approximations to the solutions of the parabolic equations with boundary integral conditions. The global extrapolation and the correction approximations, rather than the pointwise extrapolation results, are derived.
基金supported in part by the National Basic Research Program (2007CB814906)the National Natural Science Foundation of China (10471103 and 10771158)+4 种基金Social Science Foundation of the Ministry of Education of China (06JA630047)Tianjin Natural Science Foundation (07JCYBJC14300)Tianjin University of Finance and Economicssupported by the National Basic Research Program under the Grant 2005CB321701the National Natural Science Foundation of China under the Grant 10771211
文摘Asymptotic error expansions in H^1-norm for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectan- gular meshes, the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied. The higher order numerical approximations are used to generate a posteriori error estimators for the finite element approximation.
基金Acknowledgments. The author was supported by the National Natural Science Foundation of China （11101013, 11401015） and the PHR （IHLB） under Grant PHR201108074.
文摘This paper develops a new method to analyze convergence of the iterated defect correction scheme of finite element methods on rectangular grids in both two and three dimensions. The main idea is to formulate energy inner products and energy （semi）norms into matrix forms. Then, two constants of two key inequalities involved are min and max eigenvalues of two associated generalized eigenvalue problems, respectively. Local versions on the element level of these two generalized eigenvalue problems are exactly solved to obtain sharp （lower） upper bounds of these two constants. This and some essential observations for iterated solutions establish convergence in 2D and the monotone decreasing property in 3D. For two dimensions the results herein improve those in literature; for three dimensions the results herein are new. Numerical results are presented to examine theoretical results.
文摘Objective The purpose of this study is to investgate changes of cTnI in myocardial ischemic and reperfusion injury during correction of cardiac defects in children. Methods From June, 1999 to May,2000,45 children (30 male, 15 female) undergoing correction of cardiac defects were divided into three groups randomly: group Ⅰ no myocardial ischemia,group Ⅱ myocardial ischemia less than 60 minutes, group Ⅲmyocardial ischemia 】 60 minutes. There were no significant differences in the three groups in age, sex ratio, C/T ratio, or left ventricular function. Blood samples for analysis were collected before skin incision and at time intervals up to 6 days postoperatively. Analysis of creatine kinase MB.LDH and cardiac-specific troponin I was used for the detection of myocardial damage. Meantime, the ECG was checked for myocardial infarction. After the reperfusion, myocardial tissue was obtained from the free wall of right ventricle myocardial structure studies. Results The level of cTnI was increased
文摘In this paper we employ the Petrov Galerkin method for the parabolic problems to get the finite element approximate solution of high accuracy by means of the interpolation postprocessing, extrapolation and defect correction techniques.
文摘In this paper, we study an efficient asymptotically correction of a-posteriori er- ror estimator for the numerical approximation of Volterra integro-differential equations by piecewise polynomial collocation method. The deviation of the error for Volterra integro- differential equations by using the defect correction principle is defined. Also, it is shown that for m degree piecewise polynomial collocation method, our method provides O（hm＋l） as the order of the deviation of the error. The theoretical behavior is tested on examples and it is shown that the numerical results confirm the theoretical part.
基金funded by the SNF project 200020_204917 entitled"Structure preserving and fast methods for hyperbolic systems of conservation laws".
文摘We present a class of arbitrarily high order fully explicit kinetic numerical methods in compressible fluid dynamics,both in time and space,which include the relaxation schemes by Jin and Xin.These methods can use the CFL number larger or equal to unity on regular Cartesian meshes for the multi-dimensional case.These kinetic models depend on a small parameter that can be seen as a"Knudsen"number.The method is asymptotic preserving in this Knudsen number.Also,the computational costs of the method are of the same order of a fully explicit scheme.This work is the extension of Abgrall et al.(2022)[3]to multidimensional systems.We have assessed our method on several problems for two-dimensional scalar problems and Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.
文摘In this paper we introduce a Petrov-Galerkin approximation model to the solution of linear and semi-linear elliptic boundary value problems in which piecewise quadratic polynomial space and piecewise linear polynomial space are used as the shape function space and the test function space, respectively. We prove that the approximation order of the standard quadratic finite element can be attained in this Petrov-Galerkin model. Based on the so-called 'contractivity' of the interpolation operator, we further prove that the defect iterative sequence of the linear finite element solution converge to the proposed Petrov-Galerkin approximate solution.
文摘In this paper, the Wilson nonconforming finite element is considered for solving a class of second-order elliptic boundary value problems. Based on an asymptotic error expansion for the Wilson finite element, the global superconvergences, the local superconvergences and the defect correction schemes are presented.
基金This research was supported by National Science Foundation grant 19971050 and the 973 grant numberG1998030420.
文摘Focuses on a study which determined the use of the global convergences of the domain decomposition methods with Lagrangian multiplier and nonmatching grids in solving the second order elliptic boundary value problems. Background on domain decomposition and global superconvergence; Correction scheme and estimates; Numerical examples.
文摘In this paper we study the higher accuracy methods-the extrapolation and defect correction for the semidiscrete Galerkin approximations to the solutions of the parabolic equations. Theglobal extrapolation and the correction approximations of third order, rather than the pointwiseextrapolation results, are derived.
