Optimal piecewise uniform grids for singularly perturbed equations of a convection-diffusion type

Abstract

We consider the grid approximation of the Dirichlet problem for equations of an elliptic type in a strip. The higher derivatives of the equations contain the parameter e that takes arbitrary values within the half-interval (0, 1]. These convection-diffusion equations degenerate for e =0 into first-order equations. For the small values of the parameter boundary layers may occur in a neighbourhood of the part of the boundary through which the characteristics of the degenerate equations leave the domain. A special (monotonic) difference scheme on piecewise uniform grids which are refined in the s-neighbourhood of the boundary layer is used for the discretization of this problem. The scheme converges e-uniformly with rate O((N ₁¹ lnN₁+N¹ ₂ )); N₁+1 and N2 +1 are the number of nodes on a rectangular mesh across the strip and along the strip on a unit segment, respectively. It is shown that on the family of piecewise uniform grids (specified by the value s and the node density in a neighbourhood of the boundary layer) this convergence rate as to the error estimate obtained) is unimprovable in the case of the classical difference approximations of a boundary value problem. We find the grid parameters of this family, which allow us (for the same convergence rate) to decrease the error of an approximate solution. For these optimal grids the ratio of the number of nodes of the grid w1 (the grid across the strip), which are located in the s-neighbourhood of the boundary layer, to the total number of nodes of this grid tends to unity as N1 -> ∞