A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct and they determine the different scales in the solution to this problem. A Shishkin piecewise–uniform mesh is constructed, which is used, in conjunct...

We analyze finite volume schemes of arbitrary order r for the one-dimensional singularly perturbed convection-diffusion problem on the Shishkin mesh. We show that the error under the energy norm decays as (Nln(N + 1)), where 2N is the number of subintervals of the primal partition. Furthermore, at the nodal points, the error in function value approximation super-converges with order (Nln(N + 1)...

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-perconvergence rate O(N ?2 ln 2 N + N ?1:5 ln N) in a discrete-weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter. Numerical tests in...

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 O(N ?2 ln 2 N + N ?3=2) in a discrete-weighted energy norm is established under certain regularity assumption. This convergent rate is uniformly valid with respect to the singular perturbation parameter. Numerical tests indicate t...

A class of singularly perturbed two point Boundary Value Problems (BVPs) of reaction-diffusion type for third order Ordinary Differential Equations (ODEs) with a small positive parameter (ε) multiplying the highest derivative and a discontinuous source term is considered. The BVP is reduced to a weakly coupled system consisting of one first order ordinary differential equation with a suitable i...

In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting. A superconvergence rate O(N−2 ln N + N−1.5 lnN) in a discrete -weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter . Numerical tests ind...

In this article we study numerical approximation for singularly perturbed parabolic partial differential equations with time delay. A priori bounds on the exact solution and its derivatives, which are useful for the error analysis of the numerical method are given. The problem is discretized by a hybrid scheme on a generalized Shishkin mesh in spatial direction and the implicit Euler scheme on ...

In this paper, we study the stability and accuracy of adaptive finite element methods for the convection-dominated convection-diffusion-reaction problem in the twodimension space. Through various numerical examples on a type of layer-adapted grids (Shishkin grids), we show that the mesh adaptivity driven by accuracy alone cannot stabilize the scheme in all cases. Furthermore the numerical appro...

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...

Numerical approximations to the solution of a linear singularly perturbed parabolic problem are generated using a backward Euler method in time and an upwinded finite difference operator in space on a piecewise-uniform Shishkin mesh for a convectiondiffusion problem. A proof is given to show first order convergence of these numerical approximations in appropriately weighted C-norm. Numerical re...