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...

In addition, we suppose that sufficient compatibility conditions among the data of the differential equation hold, in order that the exact solution ~u ∈ C4,3(Q̄), i.e, continuity up to fourth order in space and up to third order in time. This problem is a simple model of the classical linear double–diffusion model for saturated flow in fractured porous media (Barenblatt system) developed in [1]....

The error generated by the classical upwind finite difference method on a uniform mesh, when applied to a class of singularly perturbed model ordinary differential equations with a singularly perturbed Neumann boundary condition, tends to infinity as the singular perturbation parameter tends to zero. Note that the exact solution is uniformly bounded with respect to the perturbation parameter. F...

The error generated by the classical upwind finite difference method on a uniform mesh, when applied to a class of singularly perturbed model ordinary differential equations with a singularly perturbed Neumann boundary condition, tends to infinity as the singular perturbation parameter tends to zero. Note that the exact solution is uniformly bounded with respect to the perturbation parameter. F...

The error generated by the classical upwind finite difference method on a uniform mesh, when applied to a class of singularly perturbed model ordinary differential equations with a singularly perturbed Neumann boundary condition, tends to infinity as the singular perturbation parameter tends to zero. Note that the exact solution is uniformly bounded with respect to the perturbation parameter. F...

In this paper, parameter-uniform numerical methods for a class of singularly perturbed parabolic partial differential equations with two small parameters on a rectangular domain are studied. Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. The solution is decomposed into a sum of regular and singular components. A numerical algorithm based on an upwind fini...

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...

A semilinear reaction-diffusion two-point boundary value problem, whose secondorder derivative is multiplied by a small positive parameter ε2, is considered. It can have multiple solutions. The numerical computation of solutions having interior transition layers is analysed. It is demonstrated that the accurate computation of such solutions is exceptionally difficult. To address this difficulty...

A singularly perturbed linear convection-diffusion problem for heat transfer in two dimensions with a parabolic boundary layer is solved numerically. The numerical method consists of a special piecewise uniform mesh condensing in a neighbourhood of the parabolic layer and a standard finite difference operator satisfying a discrete maximum principle. The numerical computations demonstrate numeri...