Grid Approximations of the Solution and Diffusion Flux for Singularly Perturbed Equations with Neumann Boundary Conditions

摘要

Neumann problems for singularly perturbed parabolic equations ae considered on a segment and on a rectangle #epsilon#~2. When #epsilon#=0, the parabolic equation degenerates, and only the time derivative remains. The normalized diffusion flux, i.e., the product of #epsilon# 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 convergo #epsilon# 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 #epsilon#. The error can exceed the real solution many times for small #epsilon#. For the solution of the boundary value problems allow us to approximate the solution and the normalized diffusion fluxes #epsilon#-uniformly.