Grid approximation of singularly perturbed parabolic reaction‐diffusion equations with piecewise smooth initial‐boundary conditions
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; ɛ 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.
First Published Online: 14 Oct 2010
Keyword : singularly perturbed boundary value problem, piecewise smooth initial‐boundary conditions, parabolic reaction‐diffusion equation, finite difference approximation, ϵ‐uniform convergence, compatibility conditions, special grids