In this article, a singularly perturbed convection-diffusion problem with small time lag is examined. Because of the appearance perturbation parameter, boundary layer observed in solution problem. A hybrid scheme has been constructed, which combination cubic spline method region and midpoint upwind outer on piecewise Shishkin mesh spatial direction. For discretization derivative, Crank-Nicolson...

In this article, we analyze a fully discrete $\varepsilon-$uniformly convergent finite element method for singularly perturbed convection-diffusion-reaction boundary-value problems, on piecewise-uniform meshes. Here, choose $L-$splines as basis functions. We will concentrate the convergence analysis of which employ $L-$spline functions instead their continuous counterparts. The are approximated...

A finite-element approach, based on cubic B-spline collocation, is presented for the numerical solution of Troesch’s problem. The method is used on both a uniform mesh and a piecewise-uniform Shishkin mesh, depending on the magnitude of the eigenvalues. This is due to the existence of a boundary layer at the right endpoint of the domain for relatively large eigenvalues. The problem is also solv...

In this work we consider a parabolic system of two linear singularly perturbed equations of reaction-diffusion type coupled in the reaction terms. The small values of the diffusion parameters, in general, cause that the solution has boundary layers at the ends of the spatial domain. To obtain an efficient approximation of the solution we propose a numerical method combining the Crank-Nicolson m...

This work develops an ε-uniform finite element method for singularly perturbed two-point 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 probl...