We 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.

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Abstract. A Dirichlet problem is considered for a system of two singularly perturbed parabolic reaction-diffusion equations on a rectangle. The parabolic boundary layer appears in the solution of the problem as the perturbation parameter ε tends to zero. On the basis of the decomposition solution technique, estimates for the solution and derivatives are obtained. Using the condensing mesh technique and the classical finite difference approximations of the boundary value problem under consideration, a difference scheme is constructed that converges ε-uniformly at the rate0´, where N = mins Ns, s = 1, 2, Ns + 1 and N0 + 1 are the numbers of mesh points on the axis xs and on the axis t, respectively.

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Abstract. The Dirichlet problem is considered for a quasilinear singularly perturbed parabolic convection-diffusion equation on a rectangular domain. For this problem classical finite difference (nonlinear) schemes on piecewise uniform meshes condensing in the boundary layer converge ε-uniformly at a rate that is at best first-order. Using a Richardson extrapolation technique, we construct an improved (nonlinear) scheme that is ε-uniformly convergent at the, where N and N0 define the number of nodes in the spatial and time meshes, respectively. This nonlinear scheme is used in the construction of a linearized scheme where at each time level the nonlinear term is evaluated using the computed solution from the previous time level. Furthermore, using the linearized and nonlinear improved schemes, we construct a linearized improved Richardson scheme that converges ε-uniformly at the rate O((N −1 ln N ) 2 + N −q 0 ), where q ≥ 2.

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