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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.

In some recent works, we have proposed a new approximation of stiff systems of differential equations based on an analysis of a certain error functional associated, in a natural way, with the original problem. By using standard descent schemes, the procedure can never get stuck in local minima (since the error functional only has a minimum), but will always and steadily decrease the error until getting to the original solution (the only minimum). In this paper, we propose a variable step-size implementation of our variational approach. We are able to perform this implementation without any extra functional evaluation. We show the efficacy of this new implementation on some standard problems found in the literature.

This document tries to give an overview of the different aspects that carries computer-aided ship design. It presents a little history and its evolution. It reviews the modeling, analysis and design of the ship structure. Basically, it points out the mathematical tools used in these applications.