In this paper, we discuss the parallel domain decomposition method(DDM)for solving PDE's on parallel computers. Three types of DDM: DDM with overlapping, DDM without overlapping and DDM with fictitious component a...In this paper, we discuss the parallel domain decomposition method(DDM)for solving PDE's on parallel computers. Three types of DDM: DDM with overlapping, DDM without overlapping and DDM with fictitious component are discussed in a uniform framework. The eonvergence of the asynchronous parallel algorithms based on DDM are discussed.展开更多
We extend the pure source transfer domain decomposition method(PSTDDM)to solve the perfectly matched layer approximation of Helmholtz scattering problems in heterogeneous media.We first propose some new source transfe...We extend the pure source transfer domain decomposition method(PSTDDM)to solve the perfectly matched layer approximation of Helmholtz scattering problems in heterogeneous media.We first propose some new source transfer operators,and then introduce the layer-wise and block-wise PSTDDMs based on these operators.In particular,it is proved that the solution obtained by the layer-wise PSTDDM in R2 coincides with the exact solution to the heterogeneous Helmholtz problem in the computational domain.Second,we propose the iterative layer-wise and blockwise PSTDDMs,which are designed by simply iterating the PSTDDM alternatively over two staggered decompositions of the computational domain.Finally,extensive numerical tests in two and three dimensions show that,as the preconditioner for the GMRES method,the iterative PSTDDMs are more robust and efficient than PSTDDMs for solving heterogeneous Helmholtz problems.展开更多
This paper proposes a deep-learning-based Robin-Robin domain decomposition method(DeepDDM)for Helmholtz equations.We first present the plane wave activation-based neural network(PWNN),which is more efficient for solvi...This paper proposes a deep-learning-based Robin-Robin domain decomposition method(DeepDDM)for Helmholtz equations.We first present the plane wave activation-based neural network(PWNN),which is more efficient for solving Helmholtz equations with constant coefficients and wavenumber k than finite difference methods(FDM).On this basis,we use PWNN to discretize the subproblems divided by domain decomposition methods(DDM),which is the main idea of DeepDDM.This paper will investigate the number of iterations of using DeepDDM for continuous and discontinuous Helmholtz equations.The results demonstrate that:DeepDDM exhibits behaviors consistent with conventional robust FDM-based domain decomposition method(FDM-DDM)under the same Robin parameters,i.e.,the number of iterations by DeepDDM is almost the same as that of FDM-DDM.By choosing suitable Robin parameters on different subdomains,the convergence rate is almost constant with the rise of wavenumber in both continuous and discontinuous cases.The performance of DeepDDM on Helmholtz equations may provide new insights for improving the PDE solver by deep learning.展开更多
This paper is concerned with the elliptic problems with nonlinear StefanBoltzmann boundary condition.By combining with the monotone method,the RobinRobin domain decomposition methods are proposed to decouple the nonli...This paper is concerned with the elliptic problems with nonlinear StefanBoltzmann boundary condition.By combining with the monotone method,the RobinRobin domain decomposition methods are proposed to decouple the nonlinear interface and boundary condition.The monotone properties are verified for both the multiplicative and the additive domain decomposition methods.The numerical results confirm the theoretical analysis.展开更多
To study the domain decomposition algorithms for the equations of elliptic type, the method of optimal boundary control was used to advance a new procedure for domain decomposition algorithms and regularization method...To study the domain decomposition algorithms for the equations of elliptic type, the method of optimal boundary control was used to advance a new procedure for domain decomposition algorithms and regularization method to deal with the ill posedness of the control problem. The determination of the value of the solution of the partial differential equation on the interface——the key of the domain decomposition algorithms——was transformed into a boundary control problem and the ill posedness of the control problem was overcome by regularization. The convergence of the regularizing control solution was proven and the equations which characterize the optimal control were given therefore the value of the unknown solution on the interface of the domain would be obtained by solving a series of coupling equations. Using the boundary control method the domain decomposion algorithm can be carried out.展开更多
In this paper, an absorbing Fictitious Boundary Condition (FBC) is presented to generate an iterative Domain Decomposition Method (DDM) for analyzing waveguide problems.The relaxed algorithm is introduced to improve t...In this paper, an absorbing Fictitious Boundary Condition (FBC) is presented to generate an iterative Domain Decomposition Method (DDM) for analyzing waveguide problems.The relaxed algorithm is introduced to improve the iterative convergence. And the matrix equations are solved using the multifrontal algorithm. The resulting CPU time is greatly reduced.Finally, a number of numerical examples are given to illustrate its accuracy and efficiency.展开更多
A higher-order boundary element method(HOBEM) for simulating the fully nonlinear regular wave propagation and diffraction around a fixed vertical circular cylinder is investigated. The domain decomposition method with...A higher-order boundary element method(HOBEM) for simulating the fully nonlinear regular wave propagation and diffraction around a fixed vertical circular cylinder is investigated. The domain decomposition method with continuity conditions enforced on the interfaces between the adjacent sub-domains is implemented for reducing the computational cost. By adjusting the algorithm of iterative procedure on the interfaces, four types of coupling strategies are established, that is, Dirchlet/Dirchlet-Neumman/Neumman(D/D-N/N), Dirchlet-Neumman(D-N),Neumman-Dirchlet(N-D) and Mixed Dirchlet-Neumman/Neumman-Dirchlet(Mixed D-N/N-D). Numerical simulations indicate that the domain decomposition methods can provide accurate results compared with that of the single domain method. According to the comparisons of computational efficiency, the D/D-N/N coupling strategy is recommended for the wave propagation problem. As for the wave-body interaction problem, the Mixed D-N/N-D coupling strategy can obtain the highest computational efficiency.展开更多
This paper presents a modified domain decomposition method for the numerical solution of discrete Hamilton-Jacobi-Bellman equations arising from a class of optimal control problems using diffusion models. A convergenc...This paper presents a modified domain decomposition method for the numerical solution of discrete Hamilton-Jacobi-Bellman equations arising from a class of optimal control problems using diffusion models. A convergence theorem is established. Numerical results indicate the effectiveness and accuracy of the method.展开更多
The Domain Decomposition Method(DDM) is a powerful approach to solving maily types of PDE's. DDM is especially suitable for massively Parallel computers. In the past, most research on DDM has focused on the domain...The Domain Decomposition Method(DDM) is a powerful approach to solving maily types of PDE's. DDM is especially suitable for massively Parallel computers. In the past, most research on DDM has focused on the domain splitting technique. In this paper. we focus our attention on use of a combination of techniques to solve each subproblem. The central question with DDM is that of how to doal with the pseodoboundary conditions. Here, we introduce a set of operators which act on the pseudo-boundaries in the solution process, referring to this new. procedure as the 'Generalized Domain Decomposition A.Jlethod(GDDM).' We have already obtained convergence factors for GDDM with certain classes of PDE's. These ctonvergence factors show that we can derive exact solutions of the whole problem for certain types of PDE's, and can get superior speed of convergence for other types.展开更多
An iterative nonoverlapping domain decomposition procedure is proposed and analyzed for linear elliptic problems. At the interface of two subdomains, one subdomain problem requires that Dirichlet data be passed to it ...An iterative nonoverlapping domain decomposition procedure is proposed and analyzed for linear elliptic problems. At the interface of two subdomains, one subdomain problem requires that Dirichlet data be passed to it from the previous iteration level, while the other subdomain problem requires that Neumann data be passed to it. This procedure is suitable for parallel processing. A convergence analysis is established. Standard and mixed finite element methods are employed to give discrete versions of this domain decomposition algorithm. Numerical experiments arc conducted to show the effectiveness of the method.展开更多
Corrected explicit-implicit domain decomposition(CEIDD) algorithms are studied for parallel approximation of semilinear parabolic problems on distributed memory processors. It is natural to divide the spatial domain i...Corrected explicit-implicit domain decomposition(CEIDD) algorithms are studied for parallel approximation of semilinear parabolic problems on distributed memory processors. It is natural to divide the spatial domain into some smaller parallel strips and cells using the simplest straightline interface(SI) . By using the Leray-Schauder fixed-point theorem and the discrete energy method,it is shown that the resulting CEIDD-SI algorithm is uniquely solvable,unconditionally stable and convergent. The CEIDD-SI method always suffers from the globalization of data communication when interior boundaries cross into each other inside the domain. To overcome this disadvantage,a composite interface(CI) that consists of straight segments and zigzag fractions is suggested. The corresponding CEIDD-CI algorithm is proven to be solvable,stable and convergent. Numerical experiments are presented to support the theoretical results.展开更多
A new hybrid model, which is based on domain decomposition and proposed by the authors is used for calculating the flow around a circular cylinder at low and middle Keulegan-Carpenter numbers (Kc=2～18)respectively.Th...A new hybrid model, which is based on domain decomposition and proposed by the authors is used for calculating the flow around a circular cylinder at low and middle Keulegan-Carpenter numbers (Kc=2～18)respectively.The vortex motion patterns in asymmetric regime,single pair(or transverse)regime and double pair(or diagonal)regime are successfully simulated.The calculated drag and inertial force coefficients are in better agreement with experimental data than other recent computational results.展开更多
A new numerical method based on locally modified Cartesian meshes is proposed for solving a coupled system of a fluid flow and a porous media flow.The fluid flow is modeled by the Stokes equations while the porous med...A new numerical method based on locally modified Cartesian meshes is proposed for solving a coupled system of a fluid flow and a porous media flow.The fluid flow is modeled by the Stokes equations while the porous media flow is modeled by Darcy’s law.The method is based on a Robin-Robin domain decomposition method with a Cartesian mesh with local modifications near the interface.Some computational examples are presented and discussed.展开更多
This paper is devoted to study of an iterative procedure for domain decomposition method of second order elliptic problem with mixed boundary conditions (i.e., Dirichlet condition on a part of boundary and Neumann con...This paper is devoted to study of an iterative procedure for domain decomposition method of second order elliptic problem with mixed boundary conditions (i.e., Dirichlet condition on a part of boundary and Neumann condition on the another part of boundary). For the pure Dirichlet problem, Marini and Quarteroni [3], [4] considered a similar approach, which is extended to more complex problem in this paper.展开更多
We propose a domain decomposition method for a system of quasivariational inequalities related to the HJB equation. The monotone convergence of the algorithm is also established.
In this paper,we establish a new algorithm to the non-overlapping Schwarz domain decomposition methods with changing transmission conditions for solving one dimensional advection reaction diffusion problem.More precis...In this paper,we establish a new algorithm to the non-overlapping Schwarz domain decomposition methods with changing transmission conditions for solving one dimensional advection reaction diffusion problem.More precisely,we first describe the new algorithm and prove the convergence results under several natural assumptions on the sequences of parameters which determine the transmission conditions.Then we give a simple method to estimate the new value of parameters in each iteration.The interesting advantage of our method is that one may update the better parameters in each iteration to save the computational cost for optimizing the parameters after many steps.Finally some numerical experiments are performed to show the behavior of the convergence rate for the new method.展开更多
In this paper, we present a novel technique to obtain approximate analytical solution of fractional physical models. The new technique is a combination of a domain decomposition method and natural transform method cal...In this paper, we present a novel technique to obtain approximate analytical solution of fractional physical models. The new technique is a combination of a domain decomposition method and natural transform method called a domain decomposition natural transform method (ADNTM). The fractional derivatives are considered in Caputo sense. To illustrate the power and reliability of the method some applications are provided.展开更多
The forward-backward heat equation arises in a remarkable variety of physical applications. A non-overlaping domain decomposition method was constructed to obtain numerical solutions of the forward-backward heat equa...The forward-backward heat equation arises in a remarkable variety of physical applications. A non-overlaping domain decomposition method was constructed to obtain numerical solutions of the forward-backward heat equation. The primary advantage is that the method reduces the computation time tremendously. The convergence of the given method is established. The numerical performance shows that the domain decomposition method is effective.展开更多
Large eddy simulation(LES) cooperated with a high performance parallel computing method is applied to simulate the flow in a curved duct with square cross section in the paper. The method consists of parallel domain d...Large eddy simulation(LES) cooperated with a high performance parallel computing method is applied to simulate the flow in a curved duct with square cross section in the paper. The method consists of parallel domain decomposition of grids, creation of virtual diagonal bordered matrix, assembling of boundary matrix, parallel LDL^T decomposition, parallel solving of Poisson Equation, parallel estimation of convergence and so on. The parallel computing method can solve the problems that are difficult to solve using traditional serial computing. Furthermore, existing microcomputers can be fully used to resolve some large-scale problems of complex turbulent flow.展开更多
In this paper we consider the nonoverlapping domain decomposition method based on mixed element approximation for elliptic problems in two dimentional space. We give a kind of discrete domain decomposition iterative a...In this paper we consider the nonoverlapping domain decomposition method based on mixed element approximation for elliptic problems in two dimentional space. We give a kind of discrete domain decomposition iterative algorithm using mixed finite element, the subdomain problems of which can be implemented parallelly. We also give the existence, uniqueness and convergence of the approximate solution.展开更多
基金The project supported by National Natural Science Fundation of China.
文摘In this paper, we discuss the parallel domain decomposition method(DDM)for solving PDE's on parallel computers. Three types of DDM: DDM with overlapping, DDM without overlapping and DDM with fictitious component are discussed in a uniform framework. The eonvergence of the asynchronous parallel algorithms based on DDM are discussed.
基金funded by the Natural Science Foundation of China under grants 12071401,12171238,12261160361,and 11525103the science and technology innovation Program of Hunan Province 2022RC1191.
文摘We extend the pure source transfer domain decomposition method(PSTDDM)to solve the perfectly matched layer approximation of Helmholtz scattering problems in heterogeneous media.We first propose some new source transfer operators,and then introduce the layer-wise and block-wise PSTDDMs based on these operators.In particular,it is proved that the solution obtained by the layer-wise PSTDDM in R2 coincides with the exact solution to the heterogeneous Helmholtz problem in the computational domain.Second,we propose the iterative layer-wise and blockwise PSTDDMs,which are designed by simply iterating the PSTDDM alternatively over two staggered decompositions of the computational domain.Finally,extensive numerical tests in two and three dimensions show that,as the preconditioner for the GMRES method,the iterative PSTDDMs are more robust and efficient than PSTDDMs for solving heterogeneous Helmholtz problems.
基金National Key R&D Program of China Nos.2019YFA0709600,2019YFA0709602China NSF under the grant numbers Nos.11831016,12171468,11771440,12071069+1 种基金the Fundamental Research Funds for the Central Universities(No.JGPY202101)the Innovation Foundation of Qian Xuesen Laboratory of Space Technology。
文摘This paper proposes a deep-learning-based Robin-Robin domain decomposition method(DeepDDM)for Helmholtz equations.We first present the plane wave activation-based neural network(PWNN),which is more efficient for solving Helmholtz equations with constant coefficients and wavenumber k than finite difference methods(FDM).On this basis,we use PWNN to discretize the subproblems divided by domain decomposition methods(DDM),which is the main idea of DeepDDM.This paper will investigate the number of iterations of using DeepDDM for continuous and discontinuous Helmholtz equations.The results demonstrate that:DeepDDM exhibits behaviors consistent with conventional robust FDM-based domain decomposition method(FDM-DDM)under the same Robin parameters,i.e.,the number of iterations by DeepDDM is almost the same as that of FDM-DDM.By choosing suitable Robin parameters on different subdomains,the convergence rate is almost constant with the rise of wavenumber in both continuous and discontinuous cases.The performance of DeepDDM on Helmholtz equations may provide new insights for improving the PDE solver by deep learning.
基金supported by the National Basic Research Program(2005CB321701)111 project grant(B08018)+5 种基金supported by NSFC Tianyuan Fund for Mathematics(10826105)in part by Shanghai Key Laboratory of Intelligent Information Processing(IIPL-09-003)supported by the Shanghai Natural Science Foundation(07JC14001)supported by the Global COE Programsupported in part by National 863 Program of China(2009AA012201)supported in part by Grants-in-Aid for Scientific Research(20654011,21340021)from Japan Society for the Promotion of Science.
文摘This paper is concerned with the elliptic problems with nonlinear StefanBoltzmann boundary condition.By combining with the monotone method,the RobinRobin domain decomposition methods are proposed to decouple the nonlinear interface and boundary condition.The monotone properties are verified for both the multiplicative and the additive domain decomposition methods.The numerical results confirm the theoretical analysis.
文摘To study the domain decomposition algorithms for the equations of elliptic type, the method of optimal boundary control was used to advance a new procedure for domain decomposition algorithms and regularization method to deal with the ill posedness of the control problem. The determination of the value of the solution of the partial differential equation on the interface——the key of the domain decomposition algorithms——was transformed into a boundary control problem and the ill posedness of the control problem was overcome by regularization. The convergence of the regularizing control solution was proven and the equations which characterize the optimal control were given therefore the value of the unknown solution on the interface of the domain would be obtained by solving a series of coupling equations. Using the boundary control method the domain decomposion algorithm can be carried out.
文摘In this paper, an absorbing Fictitious Boundary Condition (FBC) is presented to generate an iterative Domain Decomposition Method (DDM) for analyzing waveguide problems.The relaxed algorithm is introduced to improve the iterative convergence. And the matrix equations are solved using the multifrontal algorithm. The resulting CPU time is greatly reduced.Finally, a number of numerical examples are given to illustrate its accuracy and efficiency.
基金supported by the National Natural Science Foundation of China(Grant No.51490673)the Pre-Research Field Fund Project of the Central Military Commission of China(Grant No.61402070201)the Fundamental Research Funds for the Central Universities(Grant No.DUT18LK09)
文摘A higher-order boundary element method(HOBEM) for simulating the fully nonlinear regular wave propagation and diffraction around a fixed vertical circular cylinder is investigated. The domain decomposition method with continuity conditions enforced on the interfaces between the adjacent sub-domains is implemented for reducing the computational cost. By adjusting the algorithm of iterative procedure on the interfaces, four types of coupling strategies are established, that is, Dirchlet/Dirchlet-Neumman/Neumman(D/D-N/N), Dirchlet-Neumman(D-N),Neumman-Dirchlet(N-D) and Mixed Dirchlet-Neumman/Neumman-Dirchlet(Mixed D-N/N-D). Numerical simulations indicate that the domain decomposition methods can provide accurate results compared with that of the single domain method. According to the comparisons of computational efficiency, the D/D-N/N coupling strategy is recommended for the wave propagation problem. As for the wave-body interaction problem, the Mixed D-N/N-D coupling strategy can obtain the highest computational efficiency.
文摘This paper presents a modified domain decomposition method for the numerical solution of discrete Hamilton-Jacobi-Bellman equations arising from a class of optimal control problems using diffusion models. A convergence theorem is established. Numerical results indicate the effectiveness and accuracy of the method.
文摘The Domain Decomposition Method(DDM) is a powerful approach to solving maily types of PDE's. DDM is especially suitable for massively Parallel computers. In the past, most research on DDM has focused on the domain splitting technique. In this paper. we focus our attention on use of a combination of techniques to solve each subproblem. The central question with DDM is that of how to doal with the pseodoboundary conditions. Here, we introduce a set of operators which act on the pseudo-boundaries in the solution process, referring to this new. procedure as the 'Generalized Domain Decomposition A.Jlethod(GDDM).' We have already obtained convergence factors for GDDM with certain classes of PDE's. These ctonvergence factors show that we can derive exact solutions of the whole problem for certain types of PDE's, and can get superior speed of convergence for other types.
文摘An iterative nonoverlapping domain decomposition procedure is proposed and analyzed for linear elliptic problems. At the interface of two subdomains, one subdomain problem requires that Dirichlet data be passed to it from the previous iteration level, while the other subdomain problem requires that Neumann data be passed to it. This procedure is suitable for parallel processing. A convergence analysis is established. Standard and mixed finite element methods are employed to give discrete versions of this domain decomposition algorithm. Numerical experiments arc conducted to show the effectiveness of the method.
基金supported by National Natural Science Foundation of China (Grant No. 10871044)
文摘Corrected explicit-implicit domain decomposition(CEIDD) algorithms are studied for parallel approximation of semilinear parabolic problems on distributed memory processors. It is natural to divide the spatial domain into some smaller parallel strips and cells using the simplest straightline interface(SI) . By using the Leray-Schauder fixed-point theorem and the discrete energy method,it is shown that the resulting CEIDD-SI algorithm is uniquely solvable,unconditionally stable and convergent. The CEIDD-SI method always suffers from the globalization of data communication when interior boundaries cross into each other inside the domain. To overcome this disadvantage,a composite interface(CI) that consists of straight segments and zigzag fractions is suggested. The corresponding CEIDD-CI algorithm is proven to be solvable,stable and convergent. Numerical experiments are presented to support the theoretical results.
基金The project supported by the National Natural Science Foundations of China the LNM,Institute of Mechanics,Academia Sinica
文摘A new hybrid model, which is based on domain decomposition and proposed by the authors is used for calculating the flow around a circular cylinder at low and middle Keulegan-Carpenter numbers (Kc=2～18)respectively.The vortex motion patterns in asymmetric regime,single pair(or transverse)regime and double pair(or diagonal)regime are successfully simulated.The calculated drag and inertial force coefficients are in better agreement with experimental data than other recent computational results.
基金supported in part by the US-NIH grant R01GM096195supported by the US AFSOR grant FA9550-09-1-0520the NCSU Innovation Seed grant.
文摘A new numerical method based on locally modified Cartesian meshes is proposed for solving a coupled system of a fluid flow and a porous media flow.The fluid flow is modeled by the Stokes equations while the porous media flow is modeled by Darcy’s law.The method is based on a Robin-Robin domain decomposition method with a Cartesian mesh with local modifications near the interface.Some computational examples are presented and discussed.
文摘This paper is devoted to study of an iterative procedure for domain decomposition method of second order elliptic problem with mixed boundary conditions (i.e., Dirichlet condition on a part of boundary and Neumann condition on the another part of boundary). For the pure Dirichlet problem, Marini and Quarteroni [3], [4] considered a similar approach, which is extended to more complex problem in this paper.
基金Supported by the National Natural Science Foundation of China(No.10571046)
文摘We propose a domain decomposition method for a system of quasivariational inequalities related to the HJB equation. The monotone convergence of the algorithm is also established.
文摘In this paper,we establish a new algorithm to the non-overlapping Schwarz domain decomposition methods with changing transmission conditions for solving one dimensional advection reaction diffusion problem.More precisely,we first describe the new algorithm and prove the convergence results under several natural assumptions on the sequences of parameters which determine the transmission conditions.Then we give a simple method to estimate the new value of parameters in each iteration.The interesting advantage of our method is that one may update the better parameters in each iteration to save the computational cost for optimizing the parameters after many steps.Finally some numerical experiments are performed to show the behavior of the convergence rate for the new method.
文摘In this paper, we present a novel technique to obtain approximate analytical solution of fractional physical models. The new technique is a combination of a domain decomposition method and natural transform method called a domain decomposition natural transform method (ADNTM). The fractional derivatives are considered in Caputo sense. To illustrate the power and reliability of the method some applications are provided.
基金Supported by the Special Funds for Major State BasicResearch Projects of China (No.G19990 32 80 2 )
文摘The forward-backward heat equation arises in a remarkable variety of physical applications. A non-overlaping domain decomposition method was constructed to obtain numerical solutions of the forward-backward heat equation. The primary advantage is that the method reduces the computation time tremendously. The convergence of the given method is established. The numerical performance shows that the domain decomposition method is effective.
文摘Large eddy simulation(LES) cooperated with a high performance parallel computing method is applied to simulate the flow in a curved duct with square cross section in the paper. The method consists of parallel domain decomposition of grids, creation of virtual diagonal bordered matrix, assembling of boundary matrix, parallel LDL^T decomposition, parallel solving of Poisson Equation, parallel estimation of convergence and so on. The parallel computing method can solve the problems that are difficult to solve using traditional serial computing. Furthermore, existing microcomputers can be fully used to resolve some large-scale problems of complex turbulent flow.
文摘In this paper we consider the nonoverlapping domain decomposition method based on mixed element approximation for elliptic problems in two dimentional space. We give a kind of discrete domain decomposition iterative algorithm using mixed finite element, the subdomain problems of which can be implemented parallelly. We also give the existence, uniqueness and convergence of the approximate solution.
