Abstract: In this paper we consider a more general singular perturbation problem, that is, -εu"(x) – a(x)u'(x) + b(x)u(x) – f(x) (0 <ε<< 1) on an adaptive grid. The mesh is constructed adaptively by equidistributing a monitor function based on the arc-length of the approximated solutions. The use of equidistribution principles appears in many practical grid adaption schemes and our analysis provid insight into the convergence behaviour on such grids, and we obtain a better condition of the mesh. In addition, we determine the generation of adaptive grids using approximate method, and using maximum principle we prove the stability and uniqueness of the discrete equation. In our paper, we mainly discuss the posterior error estimates of piecewise linear interpolation and piecewise quadratic interpolant about the approximate solution. At last we derive anε-uniform error estimate for the first-order upwind discretization of the more general singular perturbation problem, and we extend the relevant results of the document to a more general case…

Key words: singular perturbation; adaptive grid; rate of convergence; error estimate

# Research on Finite Difference Method for Singular Perturbed Boundary Value Problem on Adaptive Grids

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