A, novel collocation method for a coupled system of singularly perturbed linear equations is presented. This method is based on rational spectral collocation method in barycentric form with sinh transform. By sinh tra...A, novel collocation method for a coupled system of singularly perturbed linear equations is presented. This method is based on rational spectral collocation method in barycentric form with sinh transform. By sinh transform, the original Chebyshev points are mapped into the transformed ones clustered near the singular points of the solution. The results from asymptotic analysis about the singularity solution are employed to determine the parameters in this sinh transform. Numerical experiments are carried out to demonstrate the high accuracy and efficiency of our method.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
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
We consider a singularly perturbed semilinear convection-diffusion problem with a boundary layer of attractive turning-point type. It is shown that its solution can be decomposed into a regular solution component and ...We consider a singularly perturbed semilinear convection-diffusion problem with a boundary layer of attractive turning-point type. It is shown that its solution can be decomposed into a regular solution component and a layer component. This decomposition is used to analyse the convergence of an upwinded finite difference scheme on Shishkin meshes.展开更多
基金Acknowledgments. The support from the National Natural Science Foundation of China under Grants No.10671146 and No.50678122 is acknowledged. The authors are grateful to the referee and the editor for helpful comments and suggestions.
文摘A, novel collocation method for a coupled system of singularly perturbed linear equations is presented. This method is based on rational spectral collocation method in barycentric form with sinh transform. By sinh transform, the original Chebyshev points are mapped into the transformed ones clustered near the singular points of the solution. The results from asymptotic analysis about the singularity solution are employed to determine the parameters in this sinh transform. Numerical experiments are carried out to demonstrate the high accuracy and efficiency of our method.
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.
文摘We consider a singularly perturbed semilinear convection-diffusion problem with a boundary layer of attractive turning-point type. It is shown that its solution can be decomposed into a regular solution component and a layer component. This decomposition is used to analyse the convergence of an upwinded finite difference scheme on Shishkin meshes.
