A fast adaptive numerical method for stiff two-point boundary value problems

We describe a robust, adaptive algorithm for the solution of singularly perturbed two-point boundary value problems. Many different phenomena can arise in such problems, including boundary layers, dense oscillations, and complicated or ill-conditioned internal transition regions. Working with an integral equation reformulation of the original differential equation, we introduce a method for error analysis which can be used for mesh refinement even when the solution computed on the current mesh is underresolved. Based on this method, we have constructed a black-box code for stiff problems which automatically generates an adaptive mesh resolving all features of the solution. The solver is direct and of arbitrarily high-order accuracy and requires an amount of time proportional to the number of grid points.