To solve the Poisson equation it is usually possible to discretize it into solving the corresponding linear system Ax=b.Variational quantum algorithms(VQAs)for the discretized Poisson equation have been studied before...To solve the Poisson equation it is usually possible to discretize it into solving the corresponding linear system Ax=b.Variational quantum algorithms(VQAs)for the discretized Poisson equation have been studied before.We present a VQA based on the banded Toeplitz systems for solving the Poisson equation with respect to the structural features of matrix A.In detail,we decompose the matrices A and A^(2)into a linear combination of the corresponding banded Toeplitz matrix and sparse matrices with only a few non-zero elements.For the one-dimensional Poisson equation with different boundary conditions and the d-dimensional Poisson equation with Dirichlet boundary conditions,the number of decomposition terms is less than that reported in[Phys.Rev.A 2023108,032418].Based on the decomposition of the matrix,we design quantum circuits that efficiently evaluate the cost function.Additionally,numerical simulation verifies the feasibility of the proposed algorithm.Finally,the VQAs for linear systems of equations and matrix-vector multiplications with the K-banded Toeplitz matrix T_(n)^(K)are given,where T_(n)^(K)∈R^(n×n)and K∈O(ploylogn).展开更多
A global spherical Fourier-Legendre spectral element method is proposed to solve Poisson equations and advective flow over a sphere. In the meridional direction, Legendre polynomials are used and the region is divided...A global spherical Fourier-Legendre spectral element method is proposed to solve Poisson equations and advective flow over a sphere. In the meridional direction, Legendre polynomials are used and the region is divided into several elements. In order to avoid coordinate singularities at the north and south poles in the meridional direction, Legendre-Gauss-Radau points are chosen at the elements involving the two poles. Fourier polynomials are applied in the zonal direction for its periodicity, with only one element. Then, the partial differential equations are solved on the longitude-latitude meshes without coordinate transformation between spherical and Cartesian coordinates. For verification of the proposed method, a few Poisson equations and advective flows are tested. Firstly, the method is found to be valid for test cases with smooth solution. The results of the Poisson equations demonstrate that the present method exhibits high accuracy and exponential convergence. High- precision solutions are also obtained with near negligible numerical diffusion during the time evolution for advective flow with smooth shape. Secondly, the results of advective flow with non-smooth shape and deformational flow are also shown to be reasonable and effective. As a result, the present method is proved to be capable of solving flow through different types of elements, and thereby a desirable method with reliability and high accuracy for solving partial differential equations over a sphere.展开更多
In this paper,the local fractional natural decomposition method(LFNDM)is used for solving a local fractional Poisson equation.The local fractional Poisson equation plays a significant role in the study of a potential ...In this paper,the local fractional natural decomposition method(LFNDM)is used for solving a local fractional Poisson equation.The local fractional Poisson equation plays a significant role in the study of a potential field due to a fixed electric charge or mass density distribution.Numerical examples with computer simulations are presented in this paper.The obtained results show that LFNDM is effective and convenient for application.展开更多
A new pressure Poisson equation method with viscous terms is established on staggered grids. The derivations show that the newly established pressure equation has the identical equation form in the projection method. ...A new pressure Poisson equation method with viscous terms is established on staggered grids. The derivations show that the newly established pressure equation has the identical equation form in the projection method. The results show that the two methods have the same velocity and pressure values except slight differences in the CPU time.展开更多
In this work, we investigate the solvability of the boundary value problem for the Poisson equation, involving a generalized Riemann-Liouville and the Caputo derivative of fractional order in the class of smooth funct...In this work, we investigate the solvability of the boundary value problem for the Poisson equation, involving a generalized Riemann-Liouville and the Caputo derivative of fractional order in the class of smooth functions. The considered problems are generalization of the known Dirichlet and Neumann oroblems with operators of a fractional order.展开更多
We deal with a large solution to the semilinear Poisson equation with doublepower nonlinearityΔ^(u)=u^(p)+αu^(q)in a bounded smooth domain D■R^(n),where p>1,-1<q<p andα∈R.We obtain the asymptotic behavio...We deal with a large solution to the semilinear Poisson equation with doublepower nonlinearityΔ^(u)=u^(p)+αu^(q)in a bounded smooth domain D■R^(n),where p>1,-1<q<p andα∈R.We obtain the asymptotic behavior of a solution u near the boundary OD up to the third or higher term.展开更多
A new finite difference-Chebyshev-Tau method for the solution of the two-dimensional Poisson equation is presented. Some of the numerical results are also presented which indicate that the method is satisfactory and c...A new finite difference-Chebyshev-Tau method for the solution of the two-dimensional Poisson equation is presented. Some of the numerical results are also presented which indicate that the method is satisfactory and compatible to other methods.展开更多
A new wavelet-based finite element method is proposed for solving the Poisson equation. The wavelet bases of Hermite cubic splines on the interval are employed as the multi-scale interpolation basis in the finite elem...A new wavelet-based finite element method is proposed for solving the Poisson equation. The wavelet bases of Hermite cubic splines on the interval are employed as the multi-scale interpolation basis in the finite element analysis. The lifting scheme of the wavelet-based finite element method is discussed in detail. For the orthogonal characteristics of the wavelet bases with respect to the given inner product, the corresponding multi-scale finite element equation can be decoupled across scales, totally or partially, and suited for nesting approximation. Numerical examples indicate that the proposed method has the higher efficiency and precision in solving the Poisson equation.展开更多
When using the projection method(or fractional step method)to solve the incompressible Navier-Stokes equations,the projection step involves solving a large-scale pressure Poisson equation(PPE),which is computationally...When using the projection method(or fractional step method)to solve the incompressible Navier-Stokes equations,the projection step involves solving a large-scale pressure Poisson equation(PPE),which is computationally expensive and time-consuming.In this study,a machine learning based method is proposed to solve the large-scale PPE.An machine learning(ML)-block is used to completely or partially(if not sufficiently accurate)replace the traditional PPE iterative solver thus accelerating the solution of the incompressible Navier-Stokes equations.The ML-block is designed as a multi-scale graph neural network(GNN)framework,in which the original high-resolution graph corresponds to the discrete grids of the solution domain,graphs with the same resolution are connected by graph convolution operation,and graphs with different resolutions are connected by down/up prolongation operation.The well trained MLblock will act as a general-purpose PPE solver for a certain kind of flow problems.The proposed method is verified via solving two-dimensional Kolmogorov flows(Re=1000 and Re=5000)with different source terms.On the premise of achieving a specified high precision(ML-block partially replaces the traditional iterative solver),the ML-block provides a better initial iteration value for the traditional iterative solver,which greatly reduces the number of iterations of the traditional iterative solver and speeds up the solution of the PPE.Numerical experiments show that the ML-block has great advantages in accelerating the solving of the Navier-Stokes equations while ensuring high accuracy.展开更多
P-and S-wave separation plays an important role in elastic reverse-time migration.It can reduce the artifacts caused by crosstalk between different modes and improve image quality.In addition,P-and Swave separation ca...P-and S-wave separation plays an important role in elastic reverse-time migration.It can reduce the artifacts caused by crosstalk between different modes and improve image quality.In addition,P-and Swave separation can also be used to better understand and distinguish wave types in complex media.At present,the methods for separating wave modes in anisotropic media mainly include spatial nonstationary filtering,low-rank approximation,and vector Poisson equation.Most of these methods require multiple Fourier transforms or the calculation of large matrices,which require high computational costs for problems with large scale.In this paper,an efficient method is proposed to separate the wave mode for anisotropic media by using a scalar anisotropic Poisson operator in the spatial domain.For 2D problems,the computational complexity required by this method is 1/2 of the methods based on solving a vector Poisson equation.Therefore,compared with existing methods based on pseudoHelmholtz decomposition operators,this method can significantly reduce the computational cost.Numerical examples also show that the P and S waves decomposed by this method not only have the correct amplitude and phase relative to the input wavefield but also can reduce the computational complexity significantly.展开更多
As a boundary-type meshless method, the singular hybrid boundary node method（SHBNM） is based on the modified variational principle and the moving least square（MLS） approximation, so it has the advantages of both b...As a boundary-type meshless method, the singular hybrid boundary node method（SHBNM） is based on the modified variational principle and the moving least square（MLS） approximation, so it has the advantages of both boundary element method（BEM） and meshless method. In this paper, the dual reciprocity method（DRM） is combined with SHBNM to solve Poisson equation in which the solution is divided into particular solution and general solution. The general solution is achieved by means of SHBNM, and the particular solution is approximated by using the radial basis function（RBF）. Only randomly distributed nodes on the bounding surface of the domain are required and it doesn＇t need extra equations to compute internal parameters in the domain. The postprocess is very simple. Numerical examples for the solution of Poisson equation show that high convergence rates and high accuracy with a small node number are achievable.展开更多
This work mainly focuses on the numerical solution of the Poisson equation with the Dirichlet boundary conditions. Compared to the traditional 5-point finite difference method, the Chebyshev spectral method is applied...This work mainly focuses on the numerical solution of the Poisson equation with the Dirichlet boundary conditions. Compared to the traditional 5-point finite difference method, the Chebyshev spectral method is applied. The numerical results show the Chebyshev spectral method has high accuracy and fast convergence;the more Chebyshev points are selected, the better the accuracy is. Finally, the error of two numerical results also verifies that the algorithm has high precision.展开更多
The concept of mathematical stencil and the strategy of stencil elimination for solving the finite difference equation is presented,and then a new type of the iteration algorithm is established for the Poisson equatio...The concept of mathematical stencil and the strategy of stencil elimination for solving the finite difference equation is presented,and then a new type of the iteration algorithm is established for the Poisson equation.The new algorithm has not only the obvious property of parallelism,but also faster convergence rate than that of the classical Jacobi iteration.Numerical experiments show that the time for the new algorithm is less than that of Jacobi and Gauss-Seidel methods to obtain the same precision,and the computational velocity increases obviously when the new iterative method,instead of Jacobi method,is applied to polish operation in multi-grid method,furthermore,the polynomial acceleration method is still applicable to the new iterative method.展开更多
This paper is concerned with a new method to determine the source terms P and Q of Poisson equations for grid generation, with which a satisfactory grid can be obtained. The interior grid distribution is controlled by...This paper is concerned with a new method to determine the source terms P and Q of Poisson equations for grid generation, with which a satisfactory grid can be obtained. The interior grid distribution is controlled by the prior selection of grid point distribution along the boundary of the region, and the orthogonality condition in the neighborhood of the boundary is satisfied. This grid generation technique can be widely used in the numerical solution of 2 D flow in rivers, lakes, and shallow water regions.展开更多
The XFEM(extended finite element method) has a lot of advantages over other numerical methods to resolve discontinuities across quasi-static interfaces due to the jump in fluidic parameters or surface tension.However,...The XFEM(extended finite element method) has a lot of advantages over other numerical methods to resolve discontinuities across quasi-static interfaces due to the jump in fluidic parameters or surface tension.However,singularities corresponding to enriched degrees of freedom(DOFs) embedded in XFEM arise in the discrete pressure Poisson equations.In this paper,constraints on these DOFs are derived from the interfacial equilibrium condition and introduced in terms of stabilized Lagrange multipliers designed for non-boundary-fitted meshes to address this issue.Numerical results show that the weak and strong discontinuities in pressure with straight and circular interfaces are accurately reproduced by the constraints.Comparisons with the SUPG/PSPG(streamline upwind/pressure stabilizing Petrov-Galerkin) method without Lagrange multipliers validate the applicability and flexibility of the proposed constrained algorithm to model problems with quasi-static interfaces.展开更多
In the solution domain, the inhomogeneous part of Poisson equation is approximated with the 5-order polynomial using Galerkin method, and the particular solution of the polynomial can be determined easily. Then, the s...In the solution domain, the inhomogeneous part of Poisson equation is approximated with the 5-order polynomial using Galerkin method, and the particular solution of the polynomial can be determined easily. Then, the solution of the Poisson equation is approximated by superposition of the particular solution and the Tcomplete functions related to the Laplace equation. Unknown parameters are determined by Galerkin method, so that the approximate solution is to satisfy the boundary conditions. Comparison with analogous results of others numerical method, the two calculating examples of the paper indicate that the accuracy of the method is very high, which also has a very fast convergence rate.展开更多
We study the regularity of the solution of Dirichlet problem of Poisson equations over a bounded domain.A new sufficient condition,uniformly positive reach is introduced.Under the assumption that the closure of the un...We study the regularity of the solution of Dirichlet problem of Poisson equations over a bounded domain.A new sufficient condition,uniformly positive reach is introduced.Under the assumption that the closure of the underlying domain of interest has a uniformly positive reach,the H^2 regularity of the solution of the Poisson equation is established.In particular,this includes all star-shaped domains whose closures are of positive reach,regardless if they are Lipschitz domains or non-Lipschitz domains.Application to the strong solution to the second order elliptic PDE in non-divergence form and the regularity of Helmholtz equations will be presented to demonstrate the usefulness of the new regularity condition.展开更多
We study the Poisson equation on some complete noncompact manifolds with asymptotically nonnegative curvature. We will also study the limiting behavior of the nonhomogeneous heat equation on some complete noncompact m...We study the Poisson equation on some complete noncompact manifolds with asymptotically nonnegative curvature. We will also study the limiting behavior of the nonhomogeneous heat equation on some complete noncompact manifolds with nonnegative curvature.展开更多
We present a proof of the discrete maximum principle(DMP)for the 1D Poisson equation−u"=f equipped with mixed Dirichlet-Neumann boundary conditions.The problem is discretized using finite elements of arbitrary le...We present a proof of the discrete maximum principle(DMP)for the 1D Poisson equation−u"=f equipped with mixed Dirichlet-Neumann boundary conditions.The problem is discretized using finite elements of arbitrary lengths and polynomial degrees(hp-FEM).We show that the DMP holds on all meshes with no limitations to the sizes and polynomial degrees of the elements.展开更多
The design optimization and analysis of charged particle beam systems employing intense beams requires a robust and accurate Poisson solver.This paper presents a new type of Poisson solver which allows the effects of ...The design optimization and analysis of charged particle beam systems employing intense beams requires a robust and accurate Poisson solver.This paper presents a new type of Poisson solver which allows the effects of space charge to be elegantly included into the system dynamics.This is done by casting the charge distribution function into a series of basis functions,which are then integrated with an appropriate Green’s function to find a Taylor series of the potential at a given point within the desired distribution region.In order to avoid singularities,a Duffy transformation is applied,which allows singularity-free integration and maximized convergence region when performed with the help of Differential Algebraic methods.The method is shown to perform well on the examples studied.Practical implementation choices and some of their limitations are also explored.展开更多
基金supported by the Shandong Provincial Natural Science Foundation for Quantum Science under Grant No.ZR2021LLZ002the Fundamental Research Funds for the Central Universities under Grant No.22CX03005A。
文摘To solve the Poisson equation it is usually possible to discretize it into solving the corresponding linear system Ax=b.Variational quantum algorithms(VQAs)for the discretized Poisson equation have been studied before.We present a VQA based on the banded Toeplitz systems for solving the Poisson equation with respect to the structural features of matrix A.In detail,we decompose the matrices A and A^(2)into a linear combination of the corresponding banded Toeplitz matrix and sparse matrices with only a few non-zero elements.For the one-dimensional Poisson equation with different boundary conditions and the d-dimensional Poisson equation with Dirichlet boundary conditions,the number of decomposition terms is less than that reported in[Phys.Rev.A 2023108,032418].Based on the decomposition of the matrix,we design quantum circuits that efficiently evaluate the cost function.Additionally,numerical simulation verifies the feasibility of the proposed algorithm.Finally,the VQAs for linear systems of equations and matrix-vector multiplications with the K-banded Toeplitz matrix T_(n)^(K)are given,where T_(n)^(K)∈R^(n×n)and K∈O(ploylogn).
基金supported by the Shandong Post-Doctoral Innovation Fund(Grant No.201303064)the Qingdao Post-Doctoral Application Research Project+1 种基金the National Basic Research(973) Program of China(Grant No.2012CB417402 and 2010CB950402)the National Natural Science Foundation of China(Grant No.41176017)
文摘A global spherical Fourier-Legendre spectral element method is proposed to solve Poisson equations and advective flow over a sphere. In the meridional direction, Legendre polynomials are used and the region is divided into several elements. In order to avoid coordinate singularities at the north and south poles in the meridional direction, Legendre-Gauss-Radau points are chosen at the elements involving the two poles. Fourier polynomials are applied in the zonal direction for its periodicity, with only one element. Then, the partial differential equations are solved on the longitude-latitude meshes without coordinate transformation between spherical and Cartesian coordinates. For verification of the proposed method, a few Poisson equations and advective flows are tested. Firstly, the method is found to be valid for test cases with smooth solution. The results of the Poisson equations demonstrate that the present method exhibits high accuracy and exponential convergence. High- precision solutions are also obtained with near negligible numerical diffusion during the time evolution for advective flow with smooth shape. Secondly, the results of advective flow with non-smooth shape and deformational flow are also shown to be reasonable and effective. As a result, the present method is proved to be capable of solving flow through different types of elements, and thereby a desirable method with reliability and high accuracy for solving partial differential equations over a sphere.
文摘In this paper,the local fractional natural decomposition method(LFNDM)is used for solving a local fractional Poisson equation.The local fractional Poisson equation plays a significant role in the study of a potential field due to a fixed electric charge or mass density distribution.Numerical examples with computer simulations are presented in this paper.The obtained results show that LFNDM is effective and convenient for application.
基金Project supported by the National Natural Science Foundation of China (No. 50876114)
文摘A new pressure Poisson equation method with viscous terms is established on staggered grids. The derivations show that the newly established pressure equation has the identical equation form in the projection method. The results show that the two methods have the same velocity and pressure values except slight differences in the CPU time.
文摘In this work, we investigate the solvability of the boundary value problem for the Poisson equation, involving a generalized Riemann-Liouville and the Caputo derivative of fractional order in the class of smooth functions. The considered problems are generalization of the known Dirichlet and Neumann oroblems with operators of a fractional order.
基金supported by the JSPS KAKENHI(JP22K03386)supported by the JST SPRING(JPMJSP2132)。
文摘We deal with a large solution to the semilinear Poisson equation with doublepower nonlinearityΔ^(u)=u^(p)+αu^(q)in a bounded smooth domain D■R^(n),where p>1,-1<q<p andα∈R.We obtain the asymptotic behavior of a solution u near the boundary OD up to the third or higher term.
文摘A new finite difference-Chebyshev-Tau method for the solution of the two-dimensional Poisson equation is presented. Some of the numerical results are also presented which indicate that the method is satisfactory and compatible to other methods.
基金supported by the National Natural Science Foundation of China (Nos. 50805028 and 50875195)the Open Foundation of the State Key Laboratory of Structural Analysis for In-dustrial Equipment (No. GZ0815)
文摘A new wavelet-based finite element method is proposed for solving the Poisson equation. The wavelet bases of Hermite cubic splines on the interval are employed as the multi-scale interpolation basis in the finite element analysis. The lifting scheme of the wavelet-based finite element method is discussed in detail. For the orthogonal characteristics of the wavelet bases with respect to the given inner product, the corresponding multi-scale finite element equation can be decoupled across scales, totally or partially, and suited for nesting approximation. Numerical examples indicate that the proposed method has the higher efficiency and precision in solving the Poisson equation.
基金This research was funded by the National Natural Science Foundation of China(Grant No.52108452)the Science Fund for Creative Research Groups of the National Natural Science Foundation of China(Grant No.51921006)the Guangdong Science and Technology Department(Grant No.2020B1212030001).
文摘When using the projection method(or fractional step method)to solve the incompressible Navier-Stokes equations,the projection step involves solving a large-scale pressure Poisson equation(PPE),which is computationally expensive and time-consuming.In this study,a machine learning based method is proposed to solve the large-scale PPE.An machine learning(ML)-block is used to completely or partially(if not sufficiently accurate)replace the traditional PPE iterative solver thus accelerating the solution of the incompressible Navier-Stokes equations.The ML-block is designed as a multi-scale graph neural network(GNN)framework,in which the original high-resolution graph corresponds to the discrete grids of the solution domain,graphs with the same resolution are connected by graph convolution operation,and graphs with different resolutions are connected by down/up prolongation operation.The well trained MLblock will act as a general-purpose PPE solver for a certain kind of flow problems.The proposed method is verified via solving two-dimensional Kolmogorov flows(Re=1000 and Re=5000)with different source terms.On the premise of achieving a specified high precision(ML-block partially replaces the traditional iterative solver),the ML-block provides a better initial iteration value for the traditional iterative solver,which greatly reduces the number of iterations of the traditional iterative solver and speeds up the solution of the PPE.Numerical experiments show that the ML-block has great advantages in accelerating the solving of the Navier-Stokes equations while ensuring high accuracy.
基金supported by the National Key R&D Program of China(No.2018YFA0702505)the project of CNOOC Limited(Grant No.CNOOC-KJ GJHXJSGG YF 2022-01)+1 种基金R&D Department of China National Petroleum Corporation(Investigations on fundamental experiments and advanced theoretical methods in geophysical prospecting application,2022DQ0604-02)NSFC(Grant Nos.U23B20159,41974142,42074129,12001311)。
文摘P-and S-wave separation plays an important role in elastic reverse-time migration.It can reduce the artifacts caused by crosstalk between different modes and improve image quality.In addition,P-and Swave separation can also be used to better understand and distinguish wave types in complex media.At present,the methods for separating wave modes in anisotropic media mainly include spatial nonstationary filtering,low-rank approximation,and vector Poisson equation.Most of these methods require multiple Fourier transforms or the calculation of large matrices,which require high computational costs for problems with large scale.In this paper,an efficient method is proposed to separate the wave mode for anisotropic media by using a scalar anisotropic Poisson operator in the spatial domain.For 2D problems,the computational complexity required by this method is 1/2 of the methods based on solving a vector Poisson equation.Therefore,compared with existing methods based on pseudoHelmholtz decomposition operators,this method can significantly reduce the computational cost.Numerical examples also show that the P and S waves decomposed by this method not only have the correct amplitude and phase relative to the input wavefield but also can reduce the computational complexity significantly.
基金Foundation item： Supported by the National Natural Science Foundation of China（50608036）
文摘As a boundary-type meshless method, the singular hybrid boundary node method（SHBNM） is based on the modified variational principle and the moving least square（MLS） approximation, so it has the advantages of both boundary element method（BEM） and meshless method. In this paper, the dual reciprocity method（DRM） is combined with SHBNM to solve Poisson equation in which the solution is divided into particular solution and general solution. The general solution is achieved by means of SHBNM, and the particular solution is approximated by using the radial basis function（RBF）. Only randomly distributed nodes on the bounding surface of the domain are required and it doesn＇t need extra equations to compute internal parameters in the domain. The postprocess is very simple. Numerical examples for the solution of Poisson equation show that high convergence rates and high accuracy with a small node number are achievable.
文摘This work mainly focuses on the numerical solution of the Poisson equation with the Dirichlet boundary conditions. Compared to the traditional 5-point finite difference method, the Chebyshev spectral method is applied. The numerical results show the Chebyshev spectral method has high accuracy and fast convergence;the more Chebyshev points are selected, the better the accuracy is. Finally, the error of two numerical results also verifies that the algorithm has high precision.
文摘The concept of mathematical stencil and the strategy of stencil elimination for solving the finite difference equation is presented,and then a new type of the iteration algorithm is established for the Poisson equation.The new algorithm has not only the obvious property of parallelism,but also faster convergence rate than that of the classical Jacobi iteration.Numerical experiments show that the time for the new algorithm is less than that of Jacobi and Gauss-Seidel methods to obtain the same precision,and the computational velocity increases obviously when the new iterative method,instead of Jacobi method,is applied to polish operation in multi-grid method,furthermore,the polynomial acceleration method is still applicable to the new iterative method.
文摘This paper is concerned with a new method to determine the source terms P and Q of Poisson equations for grid generation, with which a satisfactory grid can be obtained. The interior grid distribution is controlled by the prior selection of grid point distribution along the boundary of the region, and the orthogonality condition in the neighborhood of the boundary is satisfied. This grid generation technique can be widely used in the numerical solution of 2 D flow in rivers, lakes, and shallow water regions.
文摘The XFEM(extended finite element method) has a lot of advantages over other numerical methods to resolve discontinuities across quasi-static interfaces due to the jump in fluidic parameters or surface tension.However,singularities corresponding to enriched degrees of freedom(DOFs) embedded in XFEM arise in the discrete pressure Poisson equations.In this paper,constraints on these DOFs are derived from the interfacial equilibrium condition and introduced in terms of stabilized Lagrange multipliers designed for non-boundary-fitted meshes to address this issue.Numerical results show that the weak and strong discontinuities in pressure with straight and circular interfaces are accurately reproduced by the constraints.Comparisons with the SUPG/PSPG(streamline upwind/pressure stabilizing Petrov-Galerkin) method without Lagrange multipliers validate the applicability and flexibility of the proposed constrained algorithm to model problems with quasi-static interfaces.
文摘In the solution domain, the inhomogeneous part of Poisson equation is approximated with the 5-order polynomial using Galerkin method, and the particular solution of the polynomial can be determined easily. Then, the solution of the Poisson equation is approximated by superposition of the particular solution and the Tcomplete functions related to the Laplace equation. Unknown parameters are determined by Galerkin method, so that the approximate solution is to satisfy the boundary conditions. Comparison with analogous results of others numerical method, the two calculating examples of the paper indicate that the accuracy of the method is very high, which also has a very fast convergence rate.
基金partially supported by Simons collaboration(Grant No.246211)the National Institutes of Health(Grant No.P20GM104420)+1 种基金partially supported by Simons collaboration(Grant No.280646)the National Science Foundation under the(Grant No.DMS 1521537)
文摘We study the regularity of the solution of Dirichlet problem of Poisson equations over a bounded domain.A new sufficient condition,uniformly positive reach is introduced.Under the assumption that the closure of the underlying domain of interest has a uniformly positive reach,the H^2 regularity of the solution of the Poisson equation is established.In particular,this includes all star-shaped domains whose closures are of positive reach,regardless if they are Lipschitz domains or non-Lipschitz domains.Application to the strong solution to the second order elliptic PDE in non-divergence form and the regularity of Helmholtz equations will be presented to demonstrate the usefulness of the new regularity condition.
基金the studentship of The Chinese University of Hong Kongthe Foundation of Ji'nan University GDNSF(06025219),Guangdong,China
文摘We study the Poisson equation on some complete noncompact manifolds with asymptotically nonnegative curvature. We will also study the limiting behavior of the nonhomogeneous heat equation on some complete noncompact manifolds with nonnegative curvature.
基金the support of the Czech Science Foundation,proj-ects No.102/07/0496 and 102/05/0629the Grant Agency of the Academy of Sciences of the Czech Republic,project No.IAA100760702the Academy of Sciences of the Czech Republic,Institutional Research Plan No.AV0Z10190503。
文摘We present a proof of the discrete maximum principle(DMP)for the 1D Poisson equation−u"=f equipped with mixed Dirichlet-Neumann boundary conditions.The problem is discretized using finite elements of arbitrary lengths and polynomial degrees(hp-FEM).We show that the DMP holds on all meshes with no limitations to the sizes and polynomial degrees of the elements.
文摘The design optimization and analysis of charged particle beam systems employing intense beams requires a robust and accurate Poisson solver.This paper presents a new type of Poisson solver which allows the effects of space charge to be elegantly included into the system dynamics.This is done by casting the charge distribution function into a series of basis functions,which are then integrated with an appropriate Green’s function to find a Taylor series of the potential at a given point within the desired distribution region.In order to avoid singularities,a Duffy transformation is applied,which allows singularity-free integration and maximized convergence region when performed with the help of Differential Algebraic methods.The method is shown to perform well on the examples studied.Practical implementation choices and some of their limitations are also explored.
