A numerical method is proposed to approach the Approximate Inertial Man-ifolds(AIMs)in unsteady incompressible Navier-Stokes equations,using multilevel fi-nite element method with hierarchical basis functions.Followin...A numerical method is proposed to approach the Approximate Inertial Man-ifolds(AIMs)in unsteady incompressible Navier-Stokes equations,using multilevel fi-nite element method with hierarchical basis functions.Following AIMS,the unknown variables,velocity and pressure in the governing equations,are divided into two com-ponents,namely low modes and high modes.Then,the couplings between low modes and high modes,which are not accounted by standard Galerkin method,are consid-ered by AIMs,to improve the accuracy of the numerical results.Further,the multilevel finite element method with hierarchical basis functions is introduced to approach low modes and high modes in an efficient way.As an example,the flow around airfoil NACA0012 at different angles of attack has been simulated by the method presented,and the comparisons show that there is a good agreement between the present method and experimental results.In particular,the proposed method takes less computing time than the traditional method.As a conclusion,the present method is efficient in numer-ical analysis of fluid dynamics,especially in computing time.展开更多
The bounds for the eigenvalues of the stiffness matrices in the finite element discretization corresponding to Lu := - u' with zero boundary conditions by quadratic hierarchical basis are shown explicitly. The con...The bounds for the eigenvalues of the stiffness matrices in the finite element discretization corresponding to Lu := - u' with zero boundary conditions by quadratic hierarchical basis are shown explicitly. The condition number of the resulting system behaves like O(1/h) where h is the mesh size. We also analyze a main diagonal preconditioner of the stiffness matrix which reduces the condition number of the preconditioned system to O(1).展开更多
A sparse-grid method for solving multi-dimensional backward stochastic differential equations （BSDEs） based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e....A sparse-grid method for solving multi-dimensional backward stochastic differential equations （BSDEs） based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e. the Brownian space, the conditional mathe- matical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and （adaptive） hierarchical sparse-grid interpolation. Error estimates are proved for the proposed fully-discrete scheme for multi-dimensional BSDEs with certain types of simplified generator functions. Finally, several numerical examples are provided to illustrate the accuracy and efficiency of our scheme.展开更多
基金The research is supported by the National Basic Research Program of China(973 Program,Grant No.2012CB026002)the National Natural Science Foun-dation of China(Grant No.51305355).
文摘A numerical method is proposed to approach the Approximate Inertial Man-ifolds(AIMs)in unsteady incompressible Navier-Stokes equations,using multilevel fi-nite element method with hierarchical basis functions.Following AIMS,the unknown variables,velocity and pressure in the governing equations,are divided into two com-ponents,namely low modes and high modes.Then,the couplings between low modes and high modes,which are not accounted by standard Galerkin method,are consid-ered by AIMs,to improve the accuracy of the numerical results.Further,the multilevel finite element method with hierarchical basis functions is introduced to approach low modes and high modes in an efficient way.As an example,the flow around airfoil NACA0012 at different angles of attack has been simulated by the method presented,and the comparisons show that there is a good agreement between the present method and experimental results.In particular,the proposed method takes less computing time than the traditional method.As a conclusion,the present method is efficient in numer-ical analysis of fluid dynamics,especially in computing time.
基金The paper was supported by KOSEF 1999-1-103-002-3.
文摘The bounds for the eigenvalues of the stiffness matrices in the finite element discretization corresponding to Lu := - u' with zero boundary conditions by quadratic hierarchical basis are shown explicitly. The condition number of the resulting system behaves like O(1/h) where h is the mesh size. We also analyze a main diagonal preconditioner of the stiffness matrix which reduces the condition number of the preconditioned system to O(1).
基金Acknowledgments. The first author was supported by the US Air Force Office of Scientific Research under grant FA9550-11-1-0149. The first author was also supported by the Advanced Simulation Computing Research （ASCR）, Department of Energy, through the Householder Fellowship at ORNL. The ORNL is operated by UT-Battelle, LLC, for the United States Depart-ment of Energy under Contract DE-AC05-00OR22725. The second author was supported by the US Air Force Office of Scientific Research under grant FA9550-11-1-0149. The third author was supported by the Natural Science Foundation of China under grant 11171189. The third author was also supported by the Natural Science Foundation of China under grant 91130003. The thrid author was also supported by Shandong Province Natural Science Foundation under grant ZR2001AZ002.
文摘A sparse-grid method for solving multi-dimensional backward stochastic differential equations （BSDEs） based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e. the Brownian space, the conditional mathe- matical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and （adaptive） hierarchical sparse-grid interpolation. Error estimates are proved for the proposed fully-discrete scheme for multi-dimensional BSDEs with certain types of simplified generator functions. Finally, several numerical examples are provided to illustrate the accuracy and efficiency of our scheme.
