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A uniform convergence result for a turning point problem. (English) Zbl 0685.65073
Boundary and interior layers - computational and asymptotic methods, Proc. 5th Int. Conf., BAIL-V, Shanghai/China 1988, Conf. Ser. 12, 127-132 (1988).
[For the entire collection see Zbl 0678.00028.]
This paper considers the equation \((1)\quad -\epsilon u''- p(x)u'+q(x)u=f(x)\) where p, q and f are replaced by \(\bar p,\) \(\bar q\) and \(\bar f\) that are piecewise constant on a grid. It is shown that there are problems in the uniform approximation and that the error may be \(O(h^{\lambda})\), \(0<\lambda <1\). A better scheme is constructed using (1) in the form \(-\epsilon u''-xa(x)u'+b(x)u=f(x)\) where a, b and f are replaced by \(\bar a,\) \(\bar b\) and \(\bar f.\) This is shown to have a uniform error O(h) and an example calculation is given.
Reviewer: B.Burrows

65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34E15 Singular perturbations, general theory for ordinary differential equations