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...

A parameter-uniform numerical method based on Shishkin mesh is constructed and analyzed for a weakly coupled system of singularly perturbed second order reaction-diffusion equations. A B-spline collocation method is combined with a piecewise-uniform Shishkin mesh. An asymptotic error bound (independent of the perturbation parameter ε) in the maximum norm is established theoretically. To illustr...

A linear singularly perturbed interior turning point problem with a continuous convection coefficient is examined in this paper. Parameter uniform numerical methods composed of monotone finite difference operators and piecewise-uniform Shishkin meshes, are constructed and analysed for this class of problems. A refined Shishkin mesh is placed around the location of the interior layer and we cons...

In this paper, we introduce a coupled approach of local discontinuous Galerkin (LDG) and continuous finite element method (CFEM) for solving singularly perturbed convection-diffusion problems. When the coupled continuous-discontinuous linear FEM is used under the Shishkin mesh, a uniform convergence rate O(N−1 ln N) in an associated norm is established, where N is the number of elements. Numeri...

An elliptic system of M(≥ 2) singularly perturbed linear reactiondiffusion equations, coupled through their zero-order terms, is considered on the unit square. This system does not in general satisfy a maximum principle. It is solved numerically using a standard difference scheme on tensor-product Bakhvalov and Shishkin meshes. An error analysis for these numerical methods shows that one obtain...

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...

in this paper, a parameter uniform numerical method based on shishkin mesh is suggested to solve a system of second order singularly perturbed differential equations with a turning point exhibiting boundary layers. it is assumed that both equations have a turning point at the same point. an appropriate piecewise uniform mesh is considered and a classical finite difference scheme is applied on t...

Two non-uniform meshes used as part of the finite difference method to resolve singularly perturbed mixed-delay differential equations are studied in this article. The second-order derivative is multiplied by a small parameter which gives rise boundary layers at x=0 and x=3 strong interior x=1 x=2 due delay terms. We prove that almost first-order convergent on Shishkin mesh Bakhvalov–Shishkin m...