In this paper, we deal with a singularly perturbed parabolic convection-diffusion problem. Shishkin mesh and hybrid third-order finite difference scheme are adopted for the spatial discretization. Uniform backward Euler used temporal Furthermore, preconditioning approach is also to ensure uniform convergence. Numerical experiments show that method first-order accuracy in time almost space.

A semilinear reaction-diffusion equation with multiple solutions is considered in a smooth two-dimensional domain. Its diffusion parameter ε2 is arbitrarily small, which induces boundary layers. Constructing discrete suband super-solutions, we prove existence and investigate the accuracy of multiple discrete solutions on layer-adapted meshes of Bakhvalov and Shishkin types. It is shown that one...

The convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a singularly perturbed model problem in one-dimensional setting are studied. By applying the DG method with appropriately chosen numerical traces, the existence and uniqueness of the DG solution, the optimal order L2 error bounds, and 2p+1-order superconvergence of the numerical traces are established....

We consider a singularly perturbed convection-diffusion boundary value problem whose solution contains exponential and characteristic layers. The is numerically solved by the FEM SDFEM method with bilinear elements on graded mesh. For we prove almost uniform convergence superconvergence. use of mesh allows for to yield estimates in SD norm, which not possible Shishkin type meshes. Numerical res...

A nonlinear reaction-diffusion two-point boundary value problem with multiple solutions is considered. Its second-order derivative is multiplied by a small positive parameter ε, which induces boundary layers. Using dynamical systems techniques, asymptotic properties of its discrete suband super-solutions are derived. These properties are used to investigate the accuracy of solutions of a standa...

Nonlinear singularly perturbed interior layer problems are examined. Numerical results are presented for a numerical method consisting of a monotone scheme on a Shishkin mesh refined around the approximate location of the interior layer. keywords: Singular Perturbation, Shishkin mesh, Nonlinear, Interior Layer

we consider the numerical approximation of a singularly perturbed reaction-diffusion problem over a square. Two different. approaches are compared namely: adaptive isotropic mesh refinement and anisotropic mesh refinement. Thus, we compare the h-refinement and the Shishkin mesh approaches numerically with PLTMG software [l]. It is shown how isotropic elements lead to over-refinement. and how an...

In this article we propose an efficient numerical scheme based on a Shishkin mesh for a class of singularly perturbed parabolic convection-diffusion problems with boundary turning point and retarded arguments. The solution of the considered problem exhibit a boundary layer on the left side of the domain. The continuous problem is semidiscretized by means of backward Euler finite difference meth...

We examine a time dependent singularly perturbed convection-diffusion problem, where the convective coefficient contains an interior layer. A smooth transformation is introduced to align the grid to the location of the interior layer. A numerical method consisting of an upwinded finite difference operator and a piecewise-uniform Shishkin mesh is constructed in this transformed domain. Numerical...

In this article, a numerical solution is proposed for singularly perturbed delay parabolic reaction-diffusion problem with mixed-type boundary conditions. The discretized by the implicit Euler method on uniform mesh in time and extended cubic B-spline collocation Shishkin space. parameter-uniform convergence of given, it shown to be ...