On differential singular perturbation problems: a simple variational approach

Abstract

An alternative approach for the analysis and the numerical approximation of singular perturbation problems, using a variational framework, is presented. It is based on the natural idea of minimizing the residual of the differential equation measured in the $L^2$ norm. By this strategy, the approximation is based in the solution of a set of linear problems giving the descent step directions of the problem. This is the main advantage of our approach, since we can use stable and convergent methods for linear problems (without assuming the knowledge of good initial guesses used in the approximation of the associated non-linear systems necessary in Newton-type methods). Remember that for this type of problems, we should use implicit methods. We prove that our procedure can never get stuck in local minima, and the error decreases until getting to the original solution independent of the perturbation parameter. Finally, we include some numerical examples.