In this paper we establish a high order finite volume method for the fourth order singular perturbation problems.In conjunction with the optimal meshes,the numerical solutions resulting from the method have optimal co...In this paper we establish a high order finite volume method for the fourth order singular perturbation problems.In conjunction with the optimal meshes,the numerical solutions resulting from the method have optimal convergence order.Numerical experiments are presented to verify our theoretical estimates.展开更多
The paper is concerned with strongly nonlinear singularly perturbed bound- ary value problems in one dimension.The problems are solved numerically by finite- difference schemes on special meshes which are dense in the...The paper is concerned with strongly nonlinear singularly perturbed bound- ary value problems in one dimension.The problems are solved numerically by finite- difference schemes on special meshes which are dense in the boundary layers.The Bakhvalov mesh and a special piecewise equidistant mesh are analyzed.For the central scheme,error estimates are derived in a discrete L^1 norm.They are of second order and decrease together with the perturbation parameterε.The fourth-order Numerov scheme and the Shishkin mesh are also tested numerically.Numerical results showε-uniform pointwise convergence on the Bakhvalov and Shishkin meshes.展开更多
For singularly perturbed convection-diffusion problems,supercloseness analysis of the finite element method is still open on Bakhvalov-type meshes,especially in the case of 2D.The difficulties arise from the width of ...For singularly perturbed convection-diffusion problems,supercloseness analysis of the finite element method is still open on Bakhvalov-type meshes,especially in the case of 2D.The difficulties arise from the width of the mesh in the layer adjacent to the transition point,resulting in a suboptimal estimate for convergence.Existing analysis techniques cannot handle these difficulties well.To fill this gap,here a novel interpolation is designed delicately for the smooth part of the solution,bringing about the optimal supercloseness result of almost order 2 under an energy norm for the finite element method.Our theoretical result is uniform in the singular perturbation parameterεand is supported by the numerical experiments.展开更多
A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element m...A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the high-gradient boundary layers.展开更多
In this paper, using Lin＇s integral identity technique, we prove the optimal uniform convergence θ（Nx^-2ln^2Nx＋Ny^-2ln^2Ny） in the L^2-norm for singularly perturbed problems with parabolic layers. The error esti...In this paper, using Lin＇s integral identity technique, we prove the optimal uniform convergence θ（Nx^-2ln^2Nx＋Ny^-2ln^2Ny） in the L^2-norm for singularly perturbed problems with parabolic layers. The error estimate is achieved by bilinear finite elements on a Shishkin type mesh. Here Nx and Ny are the number of elements in the x- and y-directions, respectively. Numerical results are provided supporting our theoretical analysis.展开更多
A singularly perturbed advection-diffusion two-point Robin boundary value problem whose solution has a single boundary layer is considered. Based on the piecewise linear polynomial approximation, the finite element me...A singularly perturbed advection-diffusion two-point Robin boundary value problem whose solution has a single boundary layer is considered. Based on the piecewise linear polynomial approximation, the finite element method is applied to the problem. Estimation of the error between solution and the finite element approximation is given in energy norm on shishkin-type mesh.展开更多
We consider a uniform finite difference method for nonlinear singularly perturbed multi-point boundary value problem on Shishkin mesh. The problem is discretized using integral identities, interpolating quadrature rul...We consider a uniform finite difference method for nonlinear singularly perturbed multi-point boundary value problem on Shishkin mesh. The problem is discretized using integral identities, interpolating quadrature rules, exponential basis functions and remainder terms in integral form. We show that this method is the first order convergent in the discrete maximum norm for original problem (independent of the perturbation parameter ε). To illustrate the theoretical results, we solve test problem and we also give the error distributions in the solution in Table 1 and Figures 1-3.展开更多
This paper concerns a discontinuous Galerkin(DG)method for a one-dimensional singularly perturbed problem which possesses essential characteristic of second order convection-diffusion problem after some simple transfo...This paper concerns a discontinuous Galerkin(DG)method for a one-dimensional singularly perturbed problem which possesses essential characteristic of second order convection-diffusion problem after some simple transformations.We derive an optimal convergence of the DG method for eight layer-adapted meshes in a general framework.The convergence rate is valid independent of the small parameter.Furthermore,we establish a sharper L^(2)-error estimate if the true solution has a special regular component.Numerical experiments are also given.展开更多
A numerical method based on finite difference method with variable mesh is given for self-adjoint singularly perturbed two-point boundary value problems. To obtain parameter- uniform convergence, a variable mesh is co...A numerical method based on finite difference method with variable mesh is given for self-adjoint singularly perturbed two-point boundary value problems. To obtain parameter- uniform convergence, a variable mesh is constructed, which is dense in the boundary layer region and coarse in the outer region. The uniform convergence analysis of the method is discussed. The original problem is reduced to its normal form and the reduced problem is solved by finite difference method taking variable mesh. To support the efficiency of the method, several numerical examples have been considered.展开更多
The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the sp...The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the special non-uniform grids. The uniform con vergence of this scheme is proved and some numerical examples are given.展开更多
Abstract A finite element method is proposed for the singularly perturbed reaction-diffusion problem. An optimal error bound is derived, independent of the perturbation parameter.
In this paper,a bilinear Streamline-Diffusion finite element method on Bakhvalov-Shishkin mesh for singularly perturbed convection–diffusion problem is analyzed.The method is shown to be convergent uniformly in the p...In this paper,a bilinear Streamline-Diffusion finite element method on Bakhvalov-Shishkin mesh for singularly perturbed convection–diffusion problem is analyzed.The method is shown to be convergent uniformly in the perturbation parameterǫprovided only that ∈≤N^(−1).An O(N^(−2)(lnN)^(1/2))convergent rate in a discrete streamline-diffusion norm is established under certain regularity assump-tions.Finally,through numerical experiments,we verified the theoretical results.展开更多
基金the National Natural Science Foundation of China(Grant Nos.10771224,10601070)the Guangdong Provincial Natural Science Foundation of China(Grant No.05003308)+1 种基金MOE Project of Key Research Institute of Humanities and Social Sciences at UniversitiesChina-France-Russia Mathematics Collaboration(Grant No.34000-3275100)
文摘In this paper we establish a high order finite volume method for the fourth order singular perturbation problems.In conjunction with the optimal meshes,the numerical solutions resulting from the method have optimal convergence order.Numerical experiments are presented to verify our theoretical estimates.
文摘The paper is concerned with strongly nonlinear singularly perturbed bound- ary value problems in one dimension.The problems are solved numerically by finite- difference schemes on special meshes which are dense in the boundary layers.The Bakhvalov mesh and a special piecewise equidistant mesh are analyzed.For the central scheme,error estimates are derived in a discrete L^1 norm.They are of second order and decrease together with the perturbation parameterε.The fourth-order Numerov scheme and the Shishkin mesh are also tested numerically.Numerical results showε-uniform pointwise convergence on the Bakhvalov and Shishkin meshes.
基金supported by National Natural Science Foundation of China(11771257)the Shandong Provincial Natural Science Foundation of China(ZR2023YQ002,ZR2023MA007,ZR2021MA004)。
文摘For singularly perturbed convection-diffusion problems,supercloseness analysis of the finite element method is still open on Bakhvalov-type meshes,especially in the case of 2D.The difficulties arise from the width of the mesh in the layer adjacent to the transition point,resulting in a suboptimal estimate for convergence.Existing analysis techniques cannot handle these difficulties well.To fill this gap,here a novel interpolation is designed delicately for the smooth part of the solution,bringing about the optimal supercloseness result of almost order 2 under an energy norm for the finite element method.Our theoretical result is uniform in the singular perturbation parameterεand is supported by the numerical experiments.
文摘A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the high-gradient boundary layers.
文摘In this paper, using Lin＇s integral identity technique, we prove the optimal uniform convergence θ（Nx^-2ln^2Nx＋Ny^-2ln^2Ny） in the L^2-norm for singularly perturbed problems with parabolic layers. The error estimate is achieved by bilinear finite elements on a Shishkin type mesh. Here Nx and Ny are the number of elements in the x- and y-directions, respectively. Numerical results are provided supporting our theoretical analysis.
文摘A singularly perturbed advection-diffusion two-point Robin boundary value problem whose solution has a single boundary layer is considered. Based on the piecewise linear polynomial approximation, the finite element method is applied to the problem. Estimation of the error between solution and the finite element approximation is given in energy norm on shishkin-type mesh.
文摘We consider a uniform finite difference method for nonlinear singularly perturbed multi-point boundary value problem on Shishkin mesh. The problem is discretized using integral identities, interpolating quadrature rules, exponential basis functions and remainder terms in integral form. We show that this method is the first order convergent in the discrete maximum norm for original problem (independent of the perturbation parameter ε). To illustrate the theoretical results, we solve test problem and we also give the error distributions in the solution in Table 1 and Figures 1-3.
基金Supported by the National Natural Science Foundation of China(11801396)National College Students Innovation and Entrepreneurship Training Project(202210332019Z)。
文摘This paper concerns a discontinuous Galerkin(DG)method for a one-dimensional singularly perturbed problem which possesses essential characteristic of second order convection-diffusion problem after some simple transformations.We derive an optimal convergence of the DG method for eight layer-adapted meshes in a general framework.The convergence rate is valid independent of the small parameter.Furthermore,we establish a sharper L^(2)-error estimate if the true solution has a special regular component.Numerical experiments are also given.
文摘A numerical method based on finite difference method with variable mesh is given for self-adjoint singularly perturbed two-point boundary value problems. To obtain parameter- uniform convergence, a variable mesh is constructed, which is dense in the boundary layer region and coarse in the outer region. The uniform convergence analysis of the method is discussed. The original problem is reduced to its normal form and the reduced problem is solved by finite difference method taking variable mesh. To support the efficiency of the method, several numerical examples have been considered.
文摘The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the special non-uniform grids. The uniform con vergence of this scheme is proved and some numerical examples are given.
文摘Abstract A finite element method is proposed for the singularly perturbed reaction-diffusion problem. An optimal error bound is derived, independent of the perturbation parameter.
基金supported by Zhejiang Provincial Natural Science Foundation of China(Grant No.LY15A010018)Zhejiang Provincial Department of Education(Grant No.Y201431793).
文摘In this paper,a bilinear Streamline-Diffusion finite element method on Bakhvalov-Shishkin mesh for singularly perturbed convection–diffusion problem is analyzed.The method is shown to be convergent uniformly in the perturbation parameterǫprovided only that ∈≤N^(−1).An O(N^(−2)(lnN)^(1/2))convergent rate in a discrete streamline-diffusion norm is established under certain regularity assump-tions.Finally,through numerical experiments,we verified the theoretical results.
