Grid approximation of singularly perturbed parabolic reaction‐diffusion equations with piecewise smooth initial‐boundary conditions

Abstract

A Dirichlet problem is considered for a singularly perturbed parabolic reaction–diffusion equation with piecewise smooth initial‐boundary conditions on a rectangular domain. The higher‐order derivative in the equation is multiplied by a parameter ϵ ²; ϵ ? (0, 1]. For small values of ϵ, a boundary and an interior layer arises, respectively, in a neighbourhood of the lateral part of the boundary and in a neighbourhood of the characteristic of the reduced equation passing through the point of nonsmoothness of the initial function. Using the method of special grids condensing either in a neighbourhood of the boundary layer or in neighbourhoods of the boundary and interior layers, special ϵ‐uniformly convergent difference schemes are constructed and investigated. It is shown that the convergence rate of the schemes crucially depends on the type of nonsmoothness in the initial–boundary conditions.

Key words:

• singularly perturbed boundary value problem
• piecewise smooth initial‐boundary conditions
• parabolic reaction‐diffusion equation
• finite difference approximation
• ϵ‐uniform convergence
• compatibility conditions
• special grids

Notes

This research was supported by the Netherlands Research Organisation NWO under grant No. 047.016.008, the Russian Foundation for Basic Research under grants No. 04–01–00578, 04–01–89007‐NWO_a., the Boole Centre for Research in Informatics at the National University of Ireland, Cork and the Mathematics Applications Consortium for Science and Industry in Ireland (MACSI) under the Science Foundation Ireland (SFI) mathematics initiative.