Grid approximations of the solution and diffusion flux for singularly perturbed equations with Neumann boundary conditions

Abstract

Neumann problems for singularly perturbed parabolic equations are considered on a segment and on a rectangle. The second-order derivatives are multiplyed by a small parameter ε ². When ε=0, the parabolic equation degenerates, and only the time derivative remains. The normalized diffusion flux, i.e., the product of ε and the derivative in the direction of normal, is given on the boundary. The solution of a classical discretization method on a uniform grid does not converge ε-uniformly. Moreover, we show with numerical examples that, in the case of a Neumann problem, the approximate solution and, thereupon, the discretization error may increase without bound for a vanishing ε. The error can exceed the real solution many times for small ε. For the solution of the boundary value problems new special finite difference schemes are constructed. These schemes allow us to approximate the solution and the normalized diffusion fluxes ε-uniformly.

This work was supported by the Russian Foundation of Basic Research under Grant N 95-01-00039a