The Use of Defect Correction for the Solution of Parabolic Singular Perturbation Problems

Abstract

AbstractWe construct discrete approximations for a class of singularly perturbed boundary value problems, such as the Dirichlet problem for a parabolic differential equation, for which the coefficient multiplying the highest derivatives can take an arbitrarily small value from the interval (0, 1]. Discretisation errors for classical discrete methods depend on the value of this parameter and can be of a size comparable with the solution of the original problem. We describe how to construct special discrete methods for which the accuracy of the discrete solution does not depend on the value of the parameter, but only on the number of mesh points used. Moreover, using defect correction techniques, we construct a discrete method that yields a high order of accuracy with respect to the time variable. The approximation, obtained by this special method, converges in the discrete l∞‐norm to the true solution, independent of the small parameter. For a model problem we show results for our scheme and we compare them with results obtained by the classical method.