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Sufficient conditions for the uniform convergence of a difference scheme for a singularly perturbed turning point problem

A number of results exist in the literature for singularly perturbed differential equations without turning points. In particular a number of difference schemes have been proposed that satisfy a stronger than normal convergence criteria known as uniform convergence. This guarantees that the schemes model the boundary layers well. We wish to examine whether these schemes will also be uniformly convergent, if the equation has turning points. To this end we derive sufficient conditions for uniform convergence which are satisfied not only by these schemes but by a more general class of schemes. We show that the rate of convergence is determined by a characteristic parameter of the problem which may be less than one. We confirm these theoretical results by numerical calculations.