Fitted Mesh Methods for Problems with Parabolic Boundary Layers

A Dirichlet boundary value problem for a linear parabolic differential equation is studied on a rectangular domain in the x− t plane. The coefficient of the second order space derivative is a small singular perturbation parameter, which gives rise to parabolic boundary layers on the two lateral sides of the rectangle. It is proved that a numerical method, comprising a standard finite difference operator (centred in space, implicit in time) on a fitted piecewise uniform mesh of Nx × Nt elements condensing in the boundary layers, is uniform with respect to the small parameter, in the sense that its numerical solutions converge in the maximum norm to the exact solution uniformly well for all values of the parameter in the semi-open interval (0,1]. More specifically, it is shown that the errors are bounded in the maximum norm by C((N−1 x lnNx) + N−1 t ), where C is a constant independent not only of Nx and Nt but also of the small parameter. Numerical results are presented, which validate numerically this theoretical result and show that a numerical method consisting of the same finite difference operator on a uniform mesh of Nx × Nt elements is not uniform with respect to the small parameter.