Based on an asymptotic expansion of finite element,a new extrapolation formula and extrapolation cascadic multigrid method(EXCMG)are proposed,in which the new extrapolation and quadratic interpolation are used to prov...Based on an asymptotic expansion of finite element,a new extrapolation formula and extrapolation cascadic multigrid method(EXCMG)are proposed,in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid.In the case of triple grids,the error of the new initial value is analyzed in detail.A larger scale computation is completed in PC.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
The triangular linear fnite elements on piecewise uniform grid for an elliptic problem in convex polygonal domain are discussed.Global superconvergence in discrete H1-norm and global extrapolation in discrete L2-norm ...The triangular linear fnite elements on piecewise uniform grid for an elliptic problem in convex polygonal domain are discussed.Global superconvergence in discrete H1-norm and global extrapolation in discrete L2-norm are proved.Based on these global estimates the conjugate gradient method(CG)is efective,which is applied to extrapolation cascadic multigrid method(EXCMG).The numerical experiments show that EXCMG is of the global higher accuracy for both function and gradient.展开更多
For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion(M-type),separation of the main part and the finite element projection,we prove for the first time that the...For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion(M-type),separation of the main part and the finite element projection,we prove for the first time that the asymptotic error expansions of bilinear finite element have the accuracy of O(h3)for u∈H3.Based on the obtained asymptotic error expansions for linear finite elements,extrapolation cascadic multigrid method(EXCMG)can be used to solve Robin problems effectively.Furthermore,by virtue of Richardson not only the accuracy of the approximation is improved,but also a posteriori error estimation is obtained.Finally,some numerical experiments that confirm the theoretical analysis are presented.展开更多
To solve nonlinear system of equation,F(x) = 0,a continuous Newton flow x_t(t) = V(x) =-(DF(x))^(-1)F(x),x(0) =x^0 and its mathematical properties,such as the central field,global existence and uniqueness of real root...To solve nonlinear system of equation,F(x) = 0,a continuous Newton flow x_t(t) = V(x) =-(DF(x))^(-1)F(x),x(0) =x^0 and its mathematical properties,such as the central field,global existence and uniqueness of real roots and the structure of the singular surface,are studied.We concisely introduce random Newton flow algorithm(NFA) for finding all roots,based on discrete Newton flow x^(j+1)=x^j+hV{x^j) with random initial value x^0 and h∈(0,1],and three computable quantities,g_j,d_j and K_j.The numerical experiments with dimension n=300 are provided.展开更多
基金the National Natural Science Foundation of China(Grant Nos.10771063,10571053)Doctoral Programme of National Education Ministry of China(Grant No.20050542006)Programme for New Century Excellent Talents in University(Grant No.NCET-060712)
文摘Based on an asymptotic expansion of finite element,a new extrapolation formula and extrapolation cascadic multigrid method(EXCMG)are proposed,in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid.In the case of triple grids,the error of the new initial value is analyzed in detail.A larger scale computation is completed in PC.
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supported by National Natural Science Foundation of China(Grant Nos.1130117611071067 and 11226332)+1 种基金the Research Fund for the Doctoral Program of Higher Education of China(Grant No.20120162120036)the Construct Program of the Key Discipline in Hunan Province
文摘The triangular linear fnite elements on piecewise uniform grid for an elliptic problem in convex polygonal domain are discussed.Global superconvergence in discrete H1-norm and global extrapolation in discrete L2-norm are proved.Based on these global estimates the conjugate gradient method(CG)is efective,which is applied to extrapolation cascadic multigrid method(EXCMG).The numerical experiments show that EXCMG is of the global higher accuracy for both function and gradient.
基金supported by National Natural Science Foundation of China(Grant Nos.11226332,41204082 and 11071067)the China Postdoctoral Science Foundation(Grant No.2011M501295)+1 种基金the Research Fund for the Doctoral Program of Higher Education of China(Grant No.20120162120036)the Construct Program of the Key Discipline in Hunan Province
文摘For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion(M-type),separation of the main part and the finite element projection,we prove for the first time that the asymptotic error expansions of bilinear finite element have the accuracy of O(h3)for u∈H3.Based on the obtained asymptotic error expansions for linear finite elements,extrapolation cascadic multigrid method(EXCMG)can be used to solve Robin problems effectively.Furthermore,by virtue of Richardson not only the accuracy of the approximation is improved,but also a posteriori error estimation is obtained.Finally,some numerical experiments that confirm the theoretical analysis are presented.
基金National Natural Science Foundation of China(Grant Nos. 11301176,11071067 and 11226332)
文摘To solve nonlinear system of equation,F(x) = 0,a continuous Newton flow x_t(t) = V(x) =-(DF(x))^(-1)F(x),x(0) =x^0 and its mathematical properties,such as the central field,global existence and uniqueness of real roots and the structure of the singular surface,are studied.We concisely introduce random Newton flow algorithm(NFA) for finding all roots,based on discrete Newton flow x^(j+1)=x^j+hV{x^j) with random initial value x^0 and h∈(0,1],and three computable quantities,g_j,d_j and K_j.The numerical experiments with dimension n=300 are provided.