Based on an asymptotic expansion of finite element, an extrapolation cascadic multigrid method (EXCMG) is proposed, in which the new extrapolation and quadratic interpolation are used to provide a better initial val...Based on an asymptotic expansion of finite element, an extrapolation cascadic multigrid method (EXCMG) is proposed, in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid. In the case of multiple grids, both superconvergence error in H^1-norm and the optimal error in l2-norm are analyzed. The numerical experiment shows the advantage of EXCMG in comparison with CMG.展开更多
In this paper, n-degree continuous finite element method with interpolated coefficients for nonlinear initial value problem of ordinary differential equation is introduced and analyzed. An optimal superconvergence u-u...In this paper, n-degree continuous finite element method with interpolated coefficients for nonlinear initial value problem of ordinary differential equation is introduced and analyzed. An optimal superconvergence u-uh = O(hn+2), n ≥ 2, at (n + 1)-order Lobatto points in each element respectively is proved. Finally the theoretical results are tested by a numerical example.展开更多
Based on an improved orthogonal expansion in an element, a new error expression of n-degree finite element approximation uh to two-point boundary value problem is derived, and then superconvergence of two order for bo...Based on an improved orthogonal expansion in an element, a new error expression of n-degree finite element approximation uh to two-point boundary value problem is derived, and then superconvergence of two order for both function and derivatives are obtained.展开更多
This article will combine the finite element method, the interpolated coefficient finite element method, the eigenfunction expansion method, and the search-extension method to obtain the multiple solutions for semilin...This article will combine the finite element method, the interpolated coefficient finite element method, the eigenfunction expansion method, and the search-extension method to obtain the multiple solutions for semilinear elliptic equations. This strategy not only grently reduces the expensive computation, but also is successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems with non-odd nonlinearity on some convex or nonconvex domains. Numerical solutions illustrated by their graphics for visualization will show the efficiency of the approach.展开更多
Based on the eigensystem {λj,φjof -Δ, the multiple solutions for nonlinear problem Δu+f(u)=0 in Ω,u=0 onαΩ are approximated. A new search-extension method (SEM), which consists of three steps in three level sub...Based on the eigensystem {λj,φjof -Δ, the multiple solutions for nonlinear problem Δu+f(u)=0 in Ω,u=0 onαΩ are approximated. A new search-extension method (SEM), which consists of three steps in three level subspaces, is proposed. Numerical simulations for several typical nonlinear cases, i.e. f(u)=u3,u2(u-p),u2(u2-p), are completed and some conjectures are presented.展开更多
For numerical computations of multiple solutions of the nonlinear elliptic problemΔu+ f(u)=0 inΩ, u=0 onΓ, a search-extension method (SEM) was proposed and systematically studied by the authors. This paper shal...For numerical computations of multiple solutions of the nonlinear elliptic problemΔu+ f(u)=0 inΩ, u=0 onΓ, a search-extension method (SEM) was proposed and systematically studied by the authors. This paper shall complete its theoretical analysis. It is assumed that the nonlinearity is non-convex and its solution is isolated, under some conditions the corresponding linearized problem has a unique solution. By use of the compactness of the solution family and the contradiction argument, in general conditions, the high order regularity of the solution u∈H<sup>1+α</sup>,α】0 is proved. Assume that some initial value searched by suitably many eigenbases is already fallen into the neighborhood of the isolated solution, then the optimal error estimates of its nonlinear finite element approximation are shown by the duality argument and continuation method.展开更多
Based on the eigensystem {λj, φj} of -△, the multiple solutions for nonlinear problem △u + f(u)= 0 in Ω,u = 0 on Ω are approximated. A new search-extension method (SEM) is proposed, which consists of three algor...Based on the eigensystem {λj, φj} of -△, the multiple solutions for nonlinear problem △u + f(u)= 0 in Ω,u = 0 on Ω are approximated. A new search-extension method (SEM) is proposed, which consists of three algorithms in three level subspaces.Numerical experiments for f(u)= u3 in a square and L-shape domain are presented. The results show that there exist at least 3κ - 1 distinct nonzero solutions corresponding to each κ-ple eigenvalue of -△ (Conjecture 1).展开更多
Based on an asymptotic expansion of (bi)linear finite elements, a new extrapolation formula and extrapolation cascadic multigrid method (EXCMG) are proposed. The key ingredients of the proposed methods are some ne...Based on an asymptotic expansion of (bi)linear finite elements, a new extrapolation formula and extrapolation cascadic multigrid method (EXCMG) are proposed. The key ingredients of the proposed methods are some new extrapolations and quadratic interpolations, which are used to provide better initial values on the refined grid. In the case of triple grids, the errors of the new initial values are analyzed in detail. The numerical experiments show that EXCMG has higher accuracy and efficiency.展开更多
Based on an orthogonal expansion and orthogonality correction in an element, superconvergenceat symmetric points for any degree rectangular serendipity finite element approximation to second order ellipticproblem is p...Based on an orthogonal expansion and orthogonality correction in an element, superconvergenceat symmetric points for any degree rectangular serendipity finite element approximation to second order ellipticproblem is proved, and its behaviour up to the boundary is also discussed.展开更多
基金Supported by National Natural Science Foundation of China (10771063)the Doctor Programme of the National Education Committee (20050542006)
文摘Based on an asymptotic expansion of finite element, an extrapolation cascadic multigrid method (EXCMG) is proposed, in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid. In the case of multiple grids, both superconvergence error in H^1-norm and the optimal error in l2-norm are analyzed. The numerical experiment shows the advantage of EXCMG in comparison with CMG.
基金The work was supported in part by the Special Funds of State Major Basic Research Projects (Grant No.1999032804) by scientific Research Fund of Hunan Provincial Education Department (03C508).
文摘In this paper, n-degree continuous finite element method with interpolated coefficients for nonlinear initial value problem of ordinary differential equation is introduced and analyzed. An optimal superconvergence u-uh = O(hn+2), n ≥ 2, at (n + 1)-order Lobatto points in each element respectively is proved. Finally the theoretical results are tested by a numerical example.
基金This work was supported by The Special Funds for State Major Basic Research Projects(No. G1999032804)The National Natural Science Foundation of China(Grant No.10471038)
文摘Based on an improved orthogonal expansion in an element, a new error expression of n-degree finite element approximation uh to two-point boundary value problem is derived, and then superconvergence of two order for both function and derivatives are obtained.
基金This research was supported by the National Natural Science Foundation of China (10571053)Scientific Research Fund of Hunan Provincial Education Department (0513039)the Special Funds of State Major Basic Research Projects (G1999032804)
文摘This article will combine the finite element method, the interpolated coefficient finite element method, the eigenfunction expansion method, and the search-extension method to obtain the multiple solutions for semilinear elliptic equations. This strategy not only grently reduces the expensive computation, but also is successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems with non-odd nonlinearity on some convex or nonconvex domains. Numerical solutions illustrated by their graphics for visualization will show the efficiency of the approach.
基金This work was supported by the Special Funds of State Major Basic Research Projects(Grant No.G1999032804)the National Natural Science Foundation of China(Grant No.19871027)Mathematical Tianyuan Youth Foundation of the National Natural Science Foundation of China(No.10226016).
文摘Based on the eigensystem {λj,φjof -Δ, the multiple solutions for nonlinear problem Δu+f(u)=0 in Ω,u=0 onαΩ are approximated. A new search-extension method (SEM), which consists of three steps in three level subspaces, is proposed. Numerical simulations for several typical nonlinear cases, i.e. f(u)=u3,u2(u-p),u2(u2-p), are completed and some conjectures are presented.
基金This work was supported by the National Major Basic Research Projects (Grant No. G1999032804)the National Natural Science Foundation of China (Grant No.10471038, 10571053)+1 种基金the Research Fonds for Doctor Programme (Grant No. 20050542006)Programme for New Century Excellent Talent in University (GrantNo. NCET-06-0717)
文摘For numerical computations of multiple solutions of the nonlinear elliptic problemΔu+ f(u)=0 inΩ, u=0 onΓ, a search-extension method (SEM) was proposed and systematically studied by the authors. This paper shall complete its theoretical analysis. It is assumed that the nonlinearity is non-convex and its solution is isolated, under some conditions the corresponding linearized problem has a unique solution. By use of the compactness of the solution family and the contradiction argument, in general conditions, the high order regularity of the solution u∈H<sup>1+α</sup>,α】0 is proved. Assume that some initial value searched by suitably many eigenbases is already fallen into the neighborhood of the isolated solution, then the optimal error estimates of its nonlinear finite element approximation are shown by the duality argument and continuation method.
基金This work was supported by the Special Funds of the State Major Basic Research Projects(Grant No.G1999032804)the National Natural Science Foundation of China(Grant No.10471038).
文摘Based on the eigensystem {λj, φj} of -△, the multiple solutions for nonlinear problem △u + f(u)= 0 in Ω,u = 0 on Ω are approximated. A new search-extension method (SEM) is proposed, which consists of three algorithms in three level subspaces.Numerical experiments for f(u)= u3 in a square and L-shape domain are presented. The results show that there exist at least 3κ - 1 distinct nonzero solutions corresponding to each κ-ple eigenvalue of -△ (Conjecture 1).
基金The research is supported by the National Natural Science Foundation of China (No. 11071067) and the Key Laboratory of Education Ministry.
文摘Based on an asymptotic expansion of (bi)linear finite elements, a new extrapolation formula and extrapolation cascadic multigrid method (EXCMG) are proposed. The key ingredients of the proposed methods are some new extrapolations and quadratic interpolations, which are used to provide better initial values on the refined grid. In the case of triple grids, the errors of the new initial values are analyzed in detail. The numerical experiments show that EXCMG has higher accuracy and efficiency.
基金This work was supported by the Special Funds of State Major Basic Research Projects (Grant No. G1999032804) the National Natural Science Foundation of China (Grant No. 19871027).
文摘Based on an orthogonal expansion and orthogonality correction in an element, superconvergenceat symmetric points for any degree rectangular serendipity finite element approximation to second order ellipticproblem is proved, and its behaviour up to the boundary is also discussed.