A coupled system of two singularly perturbed ordinary differential equations of first order with the prescribed initial values are considered. The leading term of each equation is multiplied by a small positive parameter and the parameters may differ. The solution exhibits overlapping layers. A Shishkin mesh is constructed. A classical finite difference scheme applied on this mesh (which is pie...

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...

In this work we are interested in the numerical approximation of 1D parabolic singularly perturbed problems of reaction–diffusion type. To approximate the multiscale solution of this problem we use a numerical scheme combining the classical backward Euler method and central differencing. The scheme is defined on some special meshes which are the tensor product of a uniform mesh in time and a sp...

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...

This paper develops a semi-analytic technique for generating smooth nonuniform grids for the numerical solution of singularly perturbed two-point boundary value problems. It is based on the usual idea of mapping a uniform grid to the desired nonuniform grid. We introduce the W-grid, which depends on the perturbation parameter 1. For problems on [0,1] with a boundary layer at one end point, the ...

A coupled system of two singularly perturbed linear reaction–diffusion two-point boundary value problems is examined. The leading term of each equation is multiplied by a small positive parameter, but these parameters may have different magnitudes. The solutions to the system have boundary layers that overlap and interact. The structure of these layers is analysed, and this leads to the constru...

In this work, a singularly perturbed two-point boundary value problem of convection-diiusion type is considered. A hp version nite element method on a strongly graded piecewise uniform mesh of Shishkin type is used to solve the model problem. With the analytic assumption of the input data, it is shown that the method converges exponentially and the convergence is uniformly valid with respect to...

In this work, the bilinear nite element method on a Shishkin mesh for convection-diiusion problems is analyzed in the two-dimensional setting. A su-perconvergent rate N ?2 ln 2 N + N ?3=2 is established on a discrete energy norm. This rate is uniformly valid with respect to the singular perturbation parameter. As a by-product, an-uniform convergence of the same order is obtained for the L 2-nor...

Abstract This paper investigates singularly perturbed parabolic partial differential equations with delay in space, and the right end plane is an integral boundary condition on a rectangular domain. A small parameter multiplied higher order derivative, which gives layers, due to term, one more layer occurs rectangle numerical method comprising standard finite difference scheme piecewise uniform...

A singularly perturbed parabolic equation of convection-diffusion type with an interior layer in the initial condition is studied. The solution is decomposed into a discontinuous regular component, a continuous outflow boundary layer component and a discontinuous interior layer component. A priori parameter-explicit bounds are derived on the derivatives of these three components. Based on these...