Discrete approximations of solutions and derivatives for a singularly perturbed parabolic convection–diffusion equation

AbstractIn this paper we consider grid approximations of a boundary value problem on a segment for a singularly perturbed parabolic convection–diffusion equation; classical finite difference approximations on piecewise uniform meshes condensing in a neighborhood of the boundary layer are used. It is necessary to find numerical approximations to both the solution and its (first-order) derivatives with errors weakly depending on the perturbation parameter ε. The approximation errors in solutions and derivatives are examined in the ρ-metric, which is adequate for capturing singular solutions of problems with boundary layers. In this metric, the errors of the solution and its derivative (∂/∂t)u(x,t) are determined by the absolute errors, and the error in the derivative (∂/∂x)u(x,t) is determined by the relative error (with respect to a majorant function for this derivative) in the boundary layer and by the absolute error outside it. In the class of meshes whose mesh size in the boundary layer does not decrease away from the boundary, it is shown that there are no meshes on which the scheme converges ε-uniformly in the ρ-metric. We establish conditions imposed on the parameters of piecewise uniform meshes under which the schemes converge in the ρ-metric almost ε-uniformly, that is, at a rate of O(ε−νN−1+N0−1), where ν>0 may be arbitrarily small; N and N0 define the numbers of mesh points in x and t, respectively