In this paper, we discuss the uniform convergence of the simple upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh for solving a singularly perturbed Robin boundary value problem, and investigate the m...In this paper, we discuss the uniform convergence of the simple upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh for solving a singularly perturbed Robin boundary value problem, and investigate the midpoint upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh to achieve better uniform convergence. The elaborate ε-uniform pointwise estimates are proved by using the comparison principle and barrier functions. The numerical experiments support the theoretical results for the schemes on the meshes.展开更多
The application of a standard Galerkin finite element method for convection-diffusion problems leads to oscillations in the discrete solution, therefore stabilization seems to be necessary. We discuss several recent s...The application of a standard Galerkin finite element method for convection-diffusion problems leads to oscillations in the discrete solution, therefore stabilization seems to be necessary. We discuss several recent stabilization methods, especially its combination with a Galerkin method on layer-adapted meshes. Supercloseness results obtained allow an improvement of the discrete solution using recovery techniques.展开更多
In this paper,a second-order singularly perturbed initial value problem is considered.A hybrid scheme which is a combination of a cubic spline and a modified midpoint upwind scheme is proposed on various types of laye...In this paper,a second-order singularly perturbed initial value problem is considered.A hybrid scheme which is a combination of a cubic spline and a modified midpoint upwind scheme is proposed on various types of layer-adapted meshes.The error bounds are established for the numerical solution and for the scaled numerical derivative in the discrete maximum norm.It is observed that the numerical solution and the scaled numerical derivative are of second-order convergence on the layer-adapted meshes irrespective of the perturbation parameter.To show the performance of the proposed method,it is applied on few test examples which are in agreement with the theoretical results.Furthermore,existing results are also compared to show the robustness of the proposed scheme.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 singularly perturbed one-dimensional convection-diffusion problem is solved numeri- cMly by the finite element method based on higher order polynomials. Numerical solutions are obtained using S-type meshes with spec...A singularly perturbed one-dimensional convection-diffusion problem is solved numeri- cMly by the finite element method based on higher order polynomials. Numerical solutions are obtained using S-type meshes with special emphasis on meshes which are graded （based on a mesh generating function） in the fine mesh region. Error estimates in the s-weighted energy norm are proved. We derive an ＇optimal＇ mesh generating function in order to min- imize the constant in the error estimate. Two layer-adapted meshes defined by a recursive formulae in the fine mesh region are also considered and a new technique for proving er- ror estimates for these meshes is presented. The aim of the paper is to emphasize the importance of using optimal meshes for higher order finite element methods. Numerical experiments support all theoretical results.展开更多
We consider an optimal control problem with an 1D singularly perturbed differential state equation.For solving such problems one uses the enhanced system of the state equation and its adjoint form.Thus,we obtain a sys...We consider an optimal control problem with an 1D singularly perturbed differential state equation.For solving such problems one uses the enhanced system of the state equation and its adjoint form.Thus,we obtain a system of two convectiondiffusion equations.Using linear finite elements on adapted grids we treat the effects of two layers arising at different boundaries of the domain.We proof uniform error estimates for this method on meshes of Shishkin type.We present numerical results supporting our analysis.展开更多
In this paper we present a first supercloseness analysis for higher-order Galerkin FEM applied to a singularly perturbed convection-diffusion problem.Using a solution decomposition and a special representation of our ...In this paper we present a first supercloseness analysis for higher-order Galerkin FEM applied to a singularly perturbed convection-diffusion problem.Using a solution decomposition and a special representation of our finite element space,we are able to prove a supercloseness property of p+1/4 in the energy norm where the polynomial order p≥3 is odd.展开更多
文摘In this paper, we discuss the uniform convergence of the simple upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh for solving a singularly perturbed Robin boundary value problem, and investigate the midpoint upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh to achieve better uniform convergence. The elaborate ε-uniform pointwise estimates are proved by using the comparison principle and barrier functions. The numerical experiments support the theoretical results for the schemes on the meshes.
文摘The application of a standard Galerkin finite element method for convection-diffusion problems leads to oscillations in the discrete solution, therefore stabilization seems to be necessary. We discuss several recent stabilization methods, especially its combination with a Galerkin method on layer-adapted meshes. Supercloseness results obtained allow an improvement of the discrete solution using recovery techniques.
基金This research work is supported by the Department of Science and Technology,Government of India Under Research Grant No.EMR/2016/005805.
文摘In this paper,a second-order singularly perturbed initial value problem is considered.A hybrid scheme which is a combination of a cubic spline and a modified midpoint upwind scheme is proposed on various types of layer-adapted meshes.The error bounds are established for the numerical solution and for the scaled numerical derivative in the discrete maximum norm.It is observed that the numerical solution and the scaled numerical derivative are of second-order convergence on the layer-adapted meshes irrespective of the perturbation parameter.To show the performance of the proposed method,it is applied on few test examples which are in agreement with the theoretical results.Furthermore,existing results are also compared to show the robustness of the proposed scheme.
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 singularly perturbed one-dimensional convection-diffusion problem is solved numeri- cMly by the finite element method based on higher order polynomials. Numerical solutions are obtained using S-type meshes with special emphasis on meshes which are graded （based on a mesh generating function） in the fine mesh region. Error estimates in the s-weighted energy norm are proved. We derive an ＇optimal＇ mesh generating function in order to min- imize the constant in the error estimate. Two layer-adapted meshes defined by a recursive formulae in the fine mesh region are also considered and a new technique for proving er- ror estimates for these meshes is presented. The aim of the paper is to emphasize the importance of using optimal meshes for higher order finite element methods. Numerical experiments support all theoretical results.
文摘We consider an optimal control problem with an 1D singularly perturbed differential state equation.For solving such problems one uses the enhanced system of the state equation and its adjoint form.Thus,we obtain a system of two convectiondiffusion equations.Using linear finite elements on adapted grids we treat the effects of two layers arising at different boundaries of the domain.We proof uniform error estimates for this method on meshes of Shishkin type.We present numerical results supporting our analysis.
文摘In this paper we present a first supercloseness analysis for higher-order Galerkin FEM applied to a singularly perturbed convection-diffusion problem.Using a solution decomposition and a special representation of our finite element space,we are able to prove a supercloseness property of p+1/4 in the energy norm where the polynomial order p≥3 is odd.
