We study interpolation of a function two variables with large gradients in regions boundary layer under the assumption that Shishkin decomposition into sum regular and components is valid for interpolated function. generalize one-dimensional cubic splines, studied earlier on Bakhvalov grids, to two-dimensional case. obtain error estimates spline interpolation, uniform small parameter.

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 ...

A singularly perturbed reaction-diffusion equation is posed in a two-dimensional L-shaped domain Ω subject to a continuous Dirchlet boundary condition. Its solutions are in the Hölder space C2/3(Ω̄) and typically exhibit boundary layers and corner singularities. The problem is discretized on a tensor-product Shishkin mesh that is further refined in a neighboorhood of the vertex of angle 3π/2. We...

In this paper a singularly perturbed reaction-diffusion partial differential equation in two space dimensions is examined. By means of an appropriate decomposition, we describe the asymptotic behaviour of the solution of problems of this kind. A central finite difference scheme is constructed for this problem which involves an appropriate Shishkin mesh. We prove that the numerical approximation...

In this work, superconvergent approximation of singularly perturbed two-point boundary value problems of reaction-diiusion type and convection-diiusion type are studied. By applying the standard nite element method on the Shishkin mesh, superconvergent error bounds of (N ?1 ln(N +1)) p+1 in a discrete energy norm are established. The error bounds are uniformly valid with respect to the singular...

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)...

We consider conforming finite element approximation of fourth-order singularly perturbed problems of reaction diffusion type. We prove superconvergence of standard C1 finite element method of degree p on a modified Shishkin mesh. In particular, a superconvergence error bound of ( N−1ln(N + 1))p in a discrete energy norm is established. The error bound is uniformly valid with respect to the sing...

In the present work, three-step Taylor Galerkin finite element method(3TGFEM) and least-squares finite element method(LSFEM) have been discussed for solving parabolic singularly perturbed problems. For singularly perturbed problems, a small parameter called singular perturbation parameter is multiplied with the highest order derivative term. As this singular perturbation parameter approaches to...

Consider the singularly perturbed linear reaction-diffusion problem −ε2Δu+ bu = f in Ω ⊂ Rd, u = 0 on ∂Ω, where d ≥ 1, the domain Ω is bounded with (when d ≥ 2) Lipschitzcontinuous boundary ∂Ω, and the parameter ε satisfies 0 < ε 1. It is argued that for this type of problem, the standard energy norm v → [ε|v|1+‖v‖0] is too weak a norm to measure adequately the errors in solutions computed by f...