We study equations in divergence form with piecewise Cαcoefficients.The domains contain corners and the discontinuity surfaces are attached to the edges of the corners.We obtain piecewise C^(1,α) estimates across th...We study equations in divergence form with piecewise Cαcoefficients.The domains contain corners and the discontinuity surfaces are attached to the edges of the corners.We obtain piecewise C^(1,α) estimates across the discontinuity surfaces and provide an example to illustrate the issue regarding the regularity at the corners.展开更多
In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart finite element approximation of second-order elliptic partial differential equations with discontinuous coefficients. Since the finite ele...In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart finite element approximation of second-order elliptic partial differential equations with discontinuous coefficients. Since the finite element spaces are nonnested, weighted intergrid transfer operators, which are stable under the weighted L2 norm, are introduced to exchange information between different meshes. By analyzing the eigenvalue distribution of the preconditioned system, we prove that except a few small eigenvalues, all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and the mesh size. As a result, we get that the convergence rate of the preconditioned conjugate gradient method is quasi-uniform with respect to the jump and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.展开更多
The multigrid V-cycle methods for adaptive finite element discretizations of two-dimensional elliptic problems with discontinuous coefficients are considered.Under the conditions that the coefficient is quasi-monotone...The multigrid V-cycle methods for adaptive finite element discretizations of two-dimensional elliptic problems with discontinuous coefficients are considered.Under the conditions that the coefficient is quasi-monotone up to a constant and the meshes are locally refined by using the newest vertex bisection algorithm,some uniform convergence results are proved for the standard multigrid V-cycle algorithm with Gauss-Seidel relaxations performed only on new nodes and their immediate neighbours.The multigrid V-cycle algorithm uses O(N)operations per iteration and is optimal.展开更多
The authors will use a method in metric geometry to show an L^(P)-estimate for gradient of the weak solutions to elliptic equations with discontinuous coefficients,even the BMO semi-norms of the coefficients are not s...The authors will use a method in metric geometry to show an L^(P)-estimate for gradient of the weak solutions to elliptic equations with discontinuous coefficients,even the BMO semi-norms of the coefficients are not small.They also extend them to the weak solutions to parabolic equations.展开更多
In this paper, a singularly perturbed Robin type boundary value problem for second-order ordinary differential equation with discontinuous convection coefficient and source term is considered. A robust-layer-resolving...In this paper, a singularly perturbed Robin type boundary value problem for second-order ordinary differential equation with discontinuous convection coefficient and source term is considered. A robust-layer-resolving numerical method is proposed. An e-uniform global error estimate for the numerical solution and also to the numerical derivative are established. Numerical results are presented, which are in agreement with the theoretical predictions.展开更多
In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered in...In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of infinit number eigenvalues. Also the asymptotic formulas for eigenvalues are found.展开更多
In this paper, local multiplicative and additive multilevel methods on adaptively refined meshes are considered for second-order elliptic problems with highly discontinuous coeffi- cients. For the multilevel-precondit...In this paper, local multiplicative and additive multilevel methods on adaptively refined meshes are considered for second-order elliptic problems with highly discontinuous coeffi- cients. For the multilevel-preconditioned system, we study the distribution of its spectrum by using the abstract Schwarz theory. It is proved that, except for a few small eigenval- ues, the spectrum of the preconditioned system is bounded quasi-uniformly with respect to the jumps of the coefficient and the mesh sizes. The convergence rate of multilevel- preconditioned conjugate gradient methods is shown to be quasi-optimal regarding the jumps and the meshes. Numerical experiments are presented to illustrate the theoretical findings.展开更多
Extrapolation cascadic multigrid(EXCMG)method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linearfinite element discretization.However,it is not trivial to g...Extrapolation cascadic multigrid(EXCMG)method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linearfinite element discretization.However,it is not trivial to generalize the vertex-centred EXCMG method to cell-centeredfinite volume(FV)methods for diffusion equations with strongly discontinuous and anisotropic coefficients,since a non-nested hierarchy of grid nodes are used in the cell-centered discretization.For cell-centered FV schemes,the vertex values(auxiliary unknowns)need to be approximated by cell-centered ones(primary unknowns).One of the novelties is to propose a new gradient transfer(GT)method of interpolating vertex unknowns with cell-centered ones,which is easy to implement and applicable to general diffusion tensors.The main novelty of this paper is to design a multigrid prolongation operator based on the GT method and splitting extrapolation method,and then propose a cell-centered EXCMG method with BiCGStab smoother for solving the large linear system resulting from linear FV discretization of diffusion equations with strongly discontinuous and anisotropic coefficients.Numerical experiments are presented to demonstrate the high efficiency of the proposed method.展开更多
In this paper we investigate the existence, uniqueness and regularity of the solution of semilinear parabolic equations with coefficients that are discontinuous across the interface, some prior estimates are obtained....In this paper we investigate the existence, uniqueness and regularity of the solution of semilinear parabolic equations with coefficients that are discontinuous across the interface, some prior estimates are obtained. A net shape of the finite elements around the singular points was designed in [7] to solve the linear elliptic problems, by means of that net, we prove that the approximate solution has the same convergence rate as that without singularity.展开更多
We study R^(d)-valued mean-field stochastic differential equations with a diffusion coefficient that varies in a discontinuous manner on the L_(p)-norm of the process.We establish the existence of a unique global stro...We study R^(d)-valued mean-field stochastic differential equations with a diffusion coefficient that varies in a discontinuous manner on the L_(p)-norm of the process.We establish the existence of a unique global strong solution in the presence of a robust drift,while also investigating scenarios where the presence of a global solution is not assured.展开更多
The authors give the first convergence proof for the Lax-Friedrichs finite differencescheme for non-convex genuinely nonlinear scalar conservation laws of the formu_t + f(k(x, t), u)_x = 0,where the coefficient k(x, t...The authors give the first convergence proof for the Lax-Friedrichs finite differencescheme for non-convex genuinely nonlinear scalar conservation laws of the formu_t + f(k(x, t), u)_x = 0,where the coefficient k(x, t) is allowed to be discontinuous along curves in the (x, t)plane. In contrast to most of the existing literature on problems with discontinuouscoefficients, here the convergence proof is not based on the singular mapping approach,but rather on the div-curl lemma (but not the Young measure) and a Lax type en-tropy estimate that is robust with respect to the regularity of k(x, t). Following [14],the authors propose a definition of entropy solution that extends the classical Kruzkovdefinition to the situation where k(x, t) is piecewise Lipschitz continuous in the (x, t)plane, and prove the stability (uniqueness) of such entropy solutions, provided that theflux function satisfies a so-called crossng condition, and that strong traces of the solu-tion exist along the curves where k(x, t) is discontinuous. It is shown that a convergentsubsequence of approximations produced by the Lax-Friedrichs scheme converges tosuch an entropy solution, implying that the entire computed sequence converges.展开更多
This paper presents and analyzes a Discrete Duality Finite Volume(DDFV)method to solve 2D diffusion problems under prescribed Robin boundary conditions.The derivation of a symmetric discrete problem is established.The...This paper presents and analyzes a Discrete Duality Finite Volume(DDFV)method to solve 2D diffusion problems under prescribed Robin boundary conditions.The derivation of a symmetric discrete problem is established.The existence and uniqueness of a solution to this discrete problem are shown via the positive definiteness of its associated matrix.We show that the discrete scheme meets the Neumann problem when the parameterα→0(and,in a sense,whenα→∞the Dirichlet problem).This work is a continuation of our work regarding the development of DDFV methods.The main innovation here is taking into account Robin’s boundary conditions.We provide a few steps of Matlab implementation and numerical tests to confirm the effectiveness of the method.展开更多
The aim of this paper is to develop a fast multigrid solver for interpolation-free finite volume (FV) discretization of anisotropic elliptic interface problems on general bounded domains that can be described as a uni...The aim of this paper is to develop a fast multigrid solver for interpolation-free finite volume (FV) discretization of anisotropic elliptic interface problems on general bounded domains that can be described as a union of blocks. We assume that the curved interface falls exactly on the boundaries of blocks. The transfinite interpolation technique is applied to generate block-wise distorted quadrilateral meshes, which can resolve the interface with fine geometric details. By an extensive study of the harmonic average point method, an interpolation-free nine-point FV scheme is then derived on such multi-block grids for anisotropic elliptic interface problems with non-homogeneous jump conditions. Moreover, for the resulting linear algebraic systems from cell-centered FV discretization, a high-order prolongation operator based fast cascadic multigrid solver is developed and shown to be robust with respect to both the problem size and the jump of the diffusion coefficients. Various non-trivial examples including four interface problems and an elliptic problem in complex domain without interface, all with tens of millions of unknowns, are provided to show that the proposed multigrid solver is dozens of times faster than the classical algebraic multigrid method as implemented in the code AMG1R5 by Stüben.展开更多
A linear convection equation with discontinuous coefficients arises in wave propagation through interfaces. An interface condition is needed at the interface to select a unique solution. An upwind scheme that builds t...A linear convection equation with discontinuous coefficients arises in wave propagation through interfaces. An interface condition is needed at the interface to select a unique solution. An upwind scheme that builds this interface condition into its numerical flux is called the immersed interface upwind scheme. An ι1-error estimate of such a scheme was first established by Wen et al. （2008）. In this paper, we provide a simple analysis on the ι1-error estimate. The main idea is to formulate the solution to the underline initial-value problem into the sum of solutions to two convection equations with constant coefficients, which can then be estimated using classical methods for the initial or boundary value problems.展开更多
In this paper,we consider the numerics of the dispersion-managed Kortewegde Vries(DM-KdV)equation for describingwave propagations in inhomogeneous media.The DM-KdV equation contains a variable dispersion map with disc...In this paper,we consider the numerics of the dispersion-managed Kortewegde Vries(DM-KdV)equation for describingwave propagations in inhomogeneous media.The DM-KdV equation contains a variable dispersion map with discontinuity,which makes the solution non-smooth in time.We formally analyze the convergence order reduction problems of some popular numerical methods including finite difference and time-splitting for solving the DM-KdV equation,where a necessary constraint on the time step has been identified.Then,two exponential-type dispersionmap integrators up to second order accuracy are derived,which are efficiently incorporatedwith the Fourier pseudospectral discretization in space,and they can converge regardless the discontinuity and the step size.Numerical comparisons show the advantage of the proposed methods with the application to solitary wave dynamics and extension to the fast&strong dispersion-management regime.展开更多
基金supported by National Natural Science Foundation of China(12061080,12161087 and 12261093)the Science and Technology Project of the Education Department of Jiangxi Province(GJJ211601)supported by National Natural Science Foundation of China(11871305).
文摘We study equations in divergence form with piecewise Cαcoefficients.The domains contain corners and the discontinuity surfaces are attached to the edges of the corners.We obtain piecewise C^(1,α) estimates across the discontinuity surfaces and provide an example to illustrate the issue regarding the regularity at the corners.
基金supported by National Natural Science Foundation of China (Grant Nos.10871100 and 11071124)
文摘In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart finite element approximation of second-order elliptic partial differential equations with discontinuous coefficients. Since the finite element spaces are nonnested, weighted intergrid transfer operators, which are stable under the weighted L2 norm, are introduced to exchange information between different meshes. By analyzing the eigenvalue distribution of the preconditioned system, we prove that except a few small eigenvalues, all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and the mesh size. As a result, we get that the convergence rate of the preconditioned conjugate gradient method is quasi-uniform with respect to the jump and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.
基金supported by the NSF of China (Grant Nos.12171238,12261160361)supported in part by the China NSF for Distinguished Young Scholars (Grant No.11725106)by the China NSF major project (Grant No.11831016).
文摘The multigrid V-cycle methods for adaptive finite element discretizations of two-dimensional elliptic problems with discontinuous coefficients are considered.Under the conditions that the coefficient is quasi-monotone up to a constant and the meshes are locally refined by using the newest vertex bisection algorithm,some uniform convergence results are proved for the standard multigrid V-cycle algorithm with Gauss-Seidel relaxations performed only on new nodes and their immediate neighbours.The multigrid V-cycle algorithm uses O(N)operations per iteration and is optimal.
基金supported by the National Key R&D Program of China(No.2021YFA1003001).
文摘The authors will use a method in metric geometry to show an L^(P)-estimate for gradient of the weak solutions to elliptic equations with discontinuous coefficients,even the BMO semi-norms of the coefficients are not small.They also extend them to the weak solutions to parabolic equations.
基金the Council of Scientific and Industrial Research,New Delhi,India for its financial support.
文摘In this paper, a singularly perturbed Robin type boundary value problem for second-order ordinary differential equation with discontinuous convection coefficient and source term is considered. A robust-layer-resolving numerical method is proposed. An e-uniform global error estimate for the numerical solution and also to the numerical derivative are established. Numerical results are presented, which are in agreement with the theoretical predictions.
文摘In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of infinit number eigenvalues. Also the asymptotic formulas for eigenvalues are found.
文摘In this paper, local multiplicative and additive multilevel methods on adaptively refined meshes are considered for second-order elliptic problems with highly discontinuous coeffi- cients. For the multilevel-preconditioned system, we study the distribution of its spectrum by using the abstract Schwarz theory. It is proved that, except for a few small eigenval- ues, the spectrum of the preconditioned system is bounded quasi-uniformly with respect to the jumps of the coefficient and the mesh sizes. The convergence rate of multilevel- preconditioned conjugate gradient methods is shown to be quasi-optimal regarding the jumps and the meshes. Numerical experiments are presented to illustrate the theoretical findings.
基金supported by Science Challenge Project(Grant No.TZ2016002)the National Natural Science Foundation of China(Grant Nos.41874086 and 11971069)+1 种基金173 Program of China(Grant No.2020-JCJQ-ZD-029)the Excellent Youth Foundation of Hunan Province of China(Grant No.2018JJ1042).
文摘Extrapolation cascadic multigrid(EXCMG)method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linearfinite element discretization.However,it is not trivial to generalize the vertex-centred EXCMG method to cell-centeredfinite volume(FV)methods for diffusion equations with strongly discontinuous and anisotropic coefficients,since a non-nested hierarchy of grid nodes are used in the cell-centered discretization.For cell-centered FV schemes,the vertex values(auxiliary unknowns)need to be approximated by cell-centered ones(primary unknowns).One of the novelties is to propose a new gradient transfer(GT)method of interpolating vertex unknowns with cell-centered ones,which is easy to implement and applicable to general diffusion tensors.The main novelty of this paper is to design a multigrid prolongation operator based on the GT method and splitting extrapolation method,and then propose a cell-centered EXCMG method with BiCGStab smoother for solving the large linear system resulting from linear FV discretization of diffusion equations with strongly discontinuous and anisotropic coefficients.Numerical experiments are presented to demonstrate the high efficiency of the proposed method.
文摘In this paper we investigate the existence, uniqueness and regularity of the solution of semilinear parabolic equations with coefficients that are discontinuous across the interface, some prior estimates are obtained. A net shape of the finite elements around the singular points was designed in [7] to solve the linear elliptic problems, by means of that net, we prove that the approximate solution has the same convergence rate as that without singularity.
文摘We study R^(d)-valued mean-field stochastic differential equations with a diffusion coefficient that varies in a discontinuous manner on the L_(p)-norm of the process.We establish the existence of a unique global strong solution in the presence of a robust drift,while also investigating scenarios where the presence of a global solution is not assured.
基金Project supported by the BeMatA Program of the Research Council of Norway and the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282
文摘The authors give the first convergence proof for the Lax-Friedrichs finite differencescheme for non-convex genuinely nonlinear scalar conservation laws of the formu_t + f(k(x, t), u)_x = 0,where the coefficient k(x, t) is allowed to be discontinuous along curves in the (x, t)plane. In contrast to most of the existing literature on problems with discontinuouscoefficients, here the convergence proof is not based on the singular mapping approach,but rather on the div-curl lemma (but not the Young measure) and a Lax type en-tropy estimate that is robust with respect to the regularity of k(x, t). Following [14],the authors propose a definition of entropy solution that extends the classical Kruzkovdefinition to the situation where k(x, t) is piecewise Lipschitz continuous in the (x, t)plane, and prove the stability (uniqueness) of such entropy solutions, provided that theflux function satisfies a so-called crossng condition, and that strong traces of the solu-tion exist along the curves where k(x, t) is discontinuous. It is shown that a convergentsubsequence of approximations produced by the Lax-Friedrichs scheme converges tosuch an entropy solution, implying that the entire computed sequence converges.
文摘This paper presents and analyzes a Discrete Duality Finite Volume(DDFV)method to solve 2D diffusion problems under prescribed Robin boundary conditions.The derivation of a symmetric discrete problem is established.The existence and uniqueness of a solution to this discrete problem are shown via the positive definiteness of its associated matrix.We show that the discrete scheme meets the Neumann problem when the parameterα→0(and,in a sense,whenα→∞the Dirichlet problem).This work is a continuation of our work regarding the development of DDFV methods.The main innovation here is taking into account Robin’s boundary conditions.We provide a few steps of Matlab implementation and numerical tests to confirm the effectiveness of the method.
基金supported by the National Natural Science Foundation of China(Grant No.42274101)X.X.Wu was supported by the Fundamental Research Funds for the Central Universities of Central South University(Grant No.2020zzts354)+2 种基金H.L.Hu was supported by the National Natural Science Foundation of China(Grant No.12071128)by the Natural Science Foundation of Hunan Province(Grant No.2021JJ30434)Z.L.Li was supported by a Simons Grant No.633724.
文摘The aim of this paper is to develop a fast multigrid solver for interpolation-free finite volume (FV) discretization of anisotropic elliptic interface problems on general bounded domains that can be described as a union of blocks. We assume that the curved interface falls exactly on the boundaries of blocks. The transfinite interpolation technique is applied to generate block-wise distorted quadrilateral meshes, which can resolve the interface with fine geometric details. By an extensive study of the harmonic average point method, an interpolation-free nine-point FV scheme is then derived on such multi-block grids for anisotropic elliptic interface problems with non-homogeneous jump conditions. Moreover, for the resulting linear algebraic systems from cell-centered FV discretization, a high-order prolongation operator based fast cascadic multigrid solver is developed and shown to be robust with respect to both the problem size and the jump of the diffusion coefficients. Various non-trivial examples including four interface problems and an elliptic problem in complex domain without interface, all with tens of millions of unknowns, are provided to show that the proposed multigrid solver is dozens of times faster than the classical algebraic multigrid method as implemented in the code AMG1R5 by Stüben.
基金supported by National Science Foundation of USA(Grant No.DMS1114546)
文摘A linear convection equation with discontinuous coefficients arises in wave propagation through interfaces. An interface condition is needed at the interface to select a unique solution. An upwind scheme that builds this interface condition into its numerical flux is called the immersed interface upwind scheme. An ι1-error estimate of such a scheme was first established by Wen et al. （2008）. In this paper, we provide a simple analysis on the ι1-error estimate. The main idea is to formulate the solution to the underline initial-value problem into the sum of solutions to two convection equations with constant coefficients, which can then be estimated using classical methods for the initial or boundary value problems.
基金supported by the National Key Research and Development Program of China(No.2020YFA0714200)the Natural Science Foundation of Hubei Province No.2019CFA007,the NSFC 11901440。
文摘In this paper,we consider the numerics of the dispersion-managed Kortewegde Vries(DM-KdV)equation for describingwave propagations in inhomogeneous media.The DM-KdV equation contains a variable dispersion map with discontinuity,which makes the solution non-smooth in time.We formally analyze the convergence order reduction problems of some popular numerical methods including finite difference and time-splitting for solving the DM-KdV equation,where a necessary constraint on the time step has been identified.Then,two exponential-type dispersionmap integrators up to second order accuracy are derived,which are efficiently incorporatedwith the Fourier pseudospectral discretization in space,and they can converge regardless the discontinuity and the step size.Numerical comparisons show the advantage of the proposed methods with the application to solitary wave dynamics and extension to the fast&strong dispersion-management regime.
