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In this paper we study the numerical behaviour of elliptic problems in which a small parameter is involved and an example concerning the computation of elastic arches is analyzed using this mathematical framework. At first, the statements of the problem and its Galerkin approximations are defined and an asymptotic analysis is performed. Then we give general conditions ensuring that a numerical scheme will converge uniformly with respect to the small parameter. Finally we study an example in computation of arches working in linear elasticity conditions. We build one finite element scheme giving a locking behaviour, and another one which does not.

The characteristics of prism coupling of finite-width beams into nonlinear waveguides composed of media with diffusive nonlinearities (thermal, etc.) are calculated by including a one-dimensional diffusion equation for the nonlocal nonlinearity. The resulting longitudinal feedback leads to bistability, and the threshold value for the minimum diffusion length varies inversely with the angular detuning.

A finite element method (P~) with numerical integration for approximating the boundary value problem -Au=2e" is considered. It is shown that the discrete problem has a solution branch (with turning point) which converges uniformely to a solution branch of the continuous problem. Error estimates are given; for example it is found that I20-2~ =0(hZ-~), e>0, where 2 o and 2 o are critical values of the parameter 2 for continuous and discrete problems.