Uniform convergence analysis of finite difference approximations on adaptive mesh for general singular perturbed problems

In this paper we consider a more general singular perturbation problem, that is, -epsilon u "(x) - a(x)u'(x) + b(x)u(x) = f(x) (0 < epsilon << 1) on an adaptive grid. The mesh is constructed adaptively by equidistributing a monitor function based on the arc-length of the approximated solutions. Our analysis provide insight into the convergence behaviour on such mesh, and the posterior error estimates of piecewise linear interpolation about the approximate solution is investigated and an epsilon-uniform error estimate for the first-order upwind discretization of general singular perturbed problem is derived at last. We extend the relevant results of the document to a more general case.