A novel overlapping domain decomposition splitting algorithm based on a CrankNicolson method is developed for the stochastic nonlinear Schrödinger equation driven by a multiplicative noise with non-periodic bound...A novel overlapping domain decomposition splitting algorithm based on a CrankNicolson method is developed for the stochastic nonlinear Schrödinger equation driven by a multiplicative noise with non-periodic boundary conditions.The proposed algorithm can significantly reduce the computational cost while maintaining the similar conservation laws.Numerical experiments are dedicated to illustrating the capability of the algorithm for different spatial dimensions,as well as the various initial conditions.In particular,we compare the performance of the overlapping domain decomposition splitting algorithm with the stochastic multi-symplectic method in[S.Jiang et al.,Commun.Comput.Phys.,14(2013),393-411]and the finite difference splitting scheme in[J.Cui et al.,J.Differ.Equ.,266(2019),5625-5663].We observe that our proposed algorithm has excellent computational efficiency and is highly competitive.It provides a useful tool for solving stochastic partial differential equations.展开更多
Based on domain decomposition, a parallel two-level finite element method for the stationary Navier-Stokes equations is proposed and analyzed. The basic idea of the method is first to solve the Navier-Stokes equations...Based on domain decomposition, a parallel two-level finite element method for the stationary Navier-Stokes equations is proposed and analyzed. The basic idea of the method is first to solve the Navier-Stokes equations on a coarse grid, then to solve the resulted residual equations in parallel on a fine grid. This method has low communication complexity. It can be implemented easily. By local a priori error estimate for finite element discretizations, error bounds of the approximate solution are derived. Numerical results are also given to illustrate the high efficiency of the method.展开更多
Since the nonconforming finite elements(NFEs)play a significant role in approximating PDE eigenvalues from below,this paper develops a new and parallel two-level preconditioned Jacobi-Davidson(PJD)method for solving t...Since the nonconforming finite elements(NFEs)play a significant role in approximating PDE eigenvalues from below,this paper develops a new and parallel two-level preconditioned Jacobi-Davidson(PJD)method for solving the large scale discrete eigenvalue problems resulting from NFE discretization of 2mth(m=1.2)order elliptic eigenvalue problems.Combining a spectral projection on the coarse space and an overlapping domain decomposition(DD),a parallel preconditioned system can be solved in each iteration.A rigorous analysis reveals that the convergence rate of our two-level PJD method is optimal and scalable.Numerical results supporting our theory are given.展开更多
Domain decomposition method(DDM)is one of the most efficient and powerful methods for solving extra-large scale and intricate electromagnetic(EM)problems,fully embodying the divide-and-conquer philosophy.It provides t...Domain decomposition method(DDM)is one of the most efficient and powerful methods for solving extra-large scale and intricate electromagnetic(EM)problems,fully embodying the divide-and-conquer philosophy.It provides the strategy of dealing with a computationally huge task that is not easy to be solved directly—dividing the task into a number of smaller ones,i.e.sub-tasks,each can be readily solved independently and employing appropriate transmission conditions(TCs)accounting for the interactions communication among these sub-tasks.This paper presents a comprehensive overview of DDM,highlighting its fundamental principles and wide-ranging applications in many diverse areas,such as very-large-scale integration circuits,antenna array radiation,and wave scattering.In the evolution of this technology,DDM has gradually manifested its remarkable power of tackling complex EM problems through its merging with Laplace,wave,Maxwell equations,as well as surface integral equations and volume integral equations.The further evolved advanced algorithms such as overlapped DDM and non-overlapped DDM are also reviewed.The efficiency of the DDMs depends strongly on the TCs of EM fields at the interface among adjacent sub-domains.The diversity of TCs in differential and integral equations generates a variety of DDMs.Due to the independence of sub-domains,the DDMs are inherently well-suited for parallel processing with high flexibility,making them particularly effective for EM full-wave simulations on distributed computers.Finally,a list of remaining challenging technical issues and future perspective on the fast-evolving field will be provided.展开更多
Based on fully overlapping domain decomposition,a parallel finite element algorithm for the unsteady Oseen equations is proposed and analyzed.In this algorithm,each processor independently computes a finite element ap...Based on fully overlapping domain decomposition,a parallel finite element algorithm for the unsteady Oseen equations is proposed and analyzed.In this algorithm,each processor independently computes a finite element approximate solution in its own subdomain by using a locally refined multiscale mesh at each time step,where conforming finite element pairs are used for the spatial discretizations and backward Euler scheme is used for the temporal discretizations,respectively.Each subproblem is defined in the entire domain with vast majority of the degrees of freedom associated with the particular subdomain that it is responsible for and hence can be solved in parallel with other subproblems using an existing sequential solver without extensive recoding.The algorithm is easy to implement and has low communication cost.Error bounds of the parallel finite element approximate solutions are estimated.Numerical experiments are also given to demonstrate the effectiveness of the algorithm.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.12171047,11971458).
文摘A novel overlapping domain decomposition splitting algorithm based on a CrankNicolson method is developed for the stochastic nonlinear Schrödinger equation driven by a multiplicative noise with non-periodic boundary conditions.The proposed algorithm can significantly reduce the computational cost while maintaining the similar conservation laws.Numerical experiments are dedicated to illustrating the capability of the algorithm for different spatial dimensions,as well as the various initial conditions.In particular,we compare the performance of the overlapping domain decomposition splitting algorithm with the stochastic multi-symplectic method in[S.Jiang et al.,Commun.Comput.Phys.,14(2013),393-411]and the finite difference splitting scheme in[J.Cui et al.,J.Differ.Equ.,266(2019),5625-5663].We observe that our proposed algorithm has excellent computational efficiency and is highly competitive.It provides a useful tool for solving stochastic partial differential equations.
基金Project supported by the National Natural Science Foundation of China(No.11001061)the Science and Technology Foundation of Guizhou Province of China(No.[2008]2123)
文摘Based on domain decomposition, a parallel two-level finite element method for the stationary Navier-Stokes equations is proposed and analyzed. The basic idea of the method is first to solve the Navier-Stokes equations on a coarse grid, then to solve the resulted residual equations in parallel on a fine grid. This method has low communication complexity. It can be implemented easily. By local a priori error estimate for finite element discretizations, error bounds of the approximate solution are derived. Numerical results are also given to illustrate the high efficiency of the method.
基金supported by the China Postdoctoral Science Foundation(No.2023M742662)supported by the National Natural Science Foundation of China(Grant Nos.12071350 and 12331015).
文摘Since the nonconforming finite elements(NFEs)play a significant role in approximating PDE eigenvalues from below,this paper develops a new and parallel two-level preconditioned Jacobi-Davidson(PJD)method for solving the large scale discrete eigenvalue problems resulting from NFE discretization of 2mth(m=1.2)order elliptic eigenvalue problems.Combining a spectral projection on the coarse space and an overlapping domain decomposition(DD),a parallel preconditioned system can be solved in each iteration.A rigorous analysis reveals that the convergence rate of our two-level PJD method is optimal and scalable.Numerical results supporting our theory are given.
基金supported by the National Natural Science Foundation of China(Grant Nos.62293492,62131008,and 62188102)the National Key Research and Development Program of China(Grant No.2024YFB2908601).
文摘Domain decomposition method(DDM)is one of the most efficient and powerful methods for solving extra-large scale and intricate electromagnetic(EM)problems,fully embodying the divide-and-conquer philosophy.It provides the strategy of dealing with a computationally huge task that is not easy to be solved directly—dividing the task into a number of smaller ones,i.e.sub-tasks,each can be readily solved independently and employing appropriate transmission conditions(TCs)accounting for the interactions communication among these sub-tasks.This paper presents a comprehensive overview of DDM,highlighting its fundamental principles and wide-ranging applications in many diverse areas,such as very-large-scale integration circuits,antenna array radiation,and wave scattering.In the evolution of this technology,DDM has gradually manifested its remarkable power of tackling complex EM problems through its merging with Laplace,wave,Maxwell equations,as well as surface integral equations and volume integral equations.The further evolved advanced algorithms such as overlapped DDM and non-overlapped DDM are also reviewed.The efficiency of the DDMs depends strongly on the TCs of EM fields at the interface among adjacent sub-domains.The diversity of TCs in differential and integral equations generates a variety of DDMs.Due to the independence of sub-domains,the DDMs are inherently well-suited for parallel processing with high flexibility,making them particularly effective for EM full-wave simulations on distributed computers.Finally,a list of remaining challenging technical issues and future perspective on the fast-evolving field will be provided.
基金supported by the Natural Science Foundation of China(No.11361016)the Basic and Frontier Explore Program of Chongqing Municipality,China(No.cstc2018jcyjAX0305)Funds for the Central Universities(No.XDJK2018B032).
文摘Based on fully overlapping domain decomposition,a parallel finite element algorithm for the unsteady Oseen equations is proposed and analyzed.In this algorithm,each processor independently computes a finite element approximate solution in its own subdomain by using a locally refined multiscale mesh at each time step,where conforming finite element pairs are used for the spatial discretizations and backward Euler scheme is used for the temporal discretizations,respectively.Each subproblem is defined in the entire domain with vast majority of the degrees of freedom associated with the particular subdomain that it is responsible for and hence can be solved in parallel with other subproblems using an existing sequential solver without extensive recoding.The algorithm is easy to implement and has low communication cost.Error bounds of the parallel finite element approximate solutions are estimated.Numerical experiments are also given to demonstrate the effectiveness of the algorithm.
