A system ofM(≥ 2) coupled singularly perturbed linear reaction-diffusion equations is considered on the unit square. Under certain hypotheses on the coupling, a maximum principle is established for the differential operator. The relationship between compatibility conditions at the corners of the square and the smoothness of the solution on the closed domain is fully described. A decomposition o...

We propose a new two-grid approach based on Bellman-Kalaba quasilinearization [6] and Axelsson [4]-Xu [30] finite element two-grid method for the solution of singularly perturbed reaction-diffusion equations. The algorithms involve solving one inexpensive problem on coarse grid and solving on fine grid one linear problem obtained by quasilinearization of the differential equation about an inter...

We analyze the convergence of the multiplicative Schwarz method applied to nonsymmetric linear algebraic systems obtained from discretizations of one-dimensional singularly perturbed convection-diffusion equations by upwind and central finite differences on a Shishkin mesh. Using the algebraic structure of the Schwarz iteration matrices we derive bounds on the infinity norm of the error that ar...

This work develops an ε-uniform finite element method for singularly perturbed boundary value problems. A surprising and remarkable observation is illustrated: By inserting one node arbitrarily in any element, the new finite element solution always intersects with the original one at fixed points, and the errors at those points converge at the same rate as regular boundary value problems (witho...

It has been observed from the authors’ numerical experiments (2007) that the Local Discontinuous Galerkin (LDG) method converges uniformly under the Shishkin mesh for singularly perturbed two-point boundary problems of the convection-diffusion type. Especially when using a piecewise polynomial space of degree k, the LDG solution achieves the optimal convergence rate k+1 under the L2-norm, and a...

In this paper, we propose and analyze a high-order uniform method for solving boundary value problems (BVPs) for singularly perturbed nonlinear delay differential equations with small shifts (delay and advance). Such types of BVPs play an important role in the modeling of various real life phenomena, such as the variational problem in control theory and in the determination of the expected time...

In this paper we consider the standard bilinear nite element method (FEM) and the corresponding streamline diiusion FEM for the singularly perturbed elliptic boundary value problem ?" (@ 2 u @x 2 + @ 2 u @y 2) ? b(x; y) ru + a (x; y)u = f (x; y) in the two space dimensions. By using the asymptotic expansion method of Vishik and Lyusternik 42] and the technique we used in 25, 26], we prove that ...

This article aims at the development and analysis of a numerical scheme for solving singularly perturbed parabolic system n reaction–diffusion equations where m (with m<n) contain perturbation parameter while rest do not it. The is based on uniform mesh in temporal variable piecewise Shishkin spatial variable, together with classical finite difference approximations. Some analytical properti...

In this paper we deal with solving robustly and efficiently one-dimensional linear parabolic singularly perturbed systems of convection-diffusion type, where the diffusion parameters can be different at each equation even they have orders magnitude. The numerical algorithm combines classical upwind finite difference scheme to discretize in space fractional implicit Euler method together an appr...