Numerical treatment of nonlocal boundary value problem with layer behaviour (Q1676399)

This paper is concerned with a particular case of boundary value (three points) problem of the form \[ \varepsilon u'(x)+ a(x)u(x) = f(x),\qquad x\in \Omega, \] \[ u(0) + \gamma u(l_{1}) = A u(l) + B, \qquad l_{1}\in \Omega, \] with the perturbation parameter \( 0<\varepsilon \ll 1\) and \(\Omega = (0,1)\). For the numerical solutions of this problem the finite difference method with two steps \(h_{1}\) and \(h_{2}\), respectively, on the intervals \((0,\sigma)\) and \((\sigma,l)\) (Shishkin mesh) is used, imposing several conditions on the data of the problem. The numerical results illustrate the behaviour of the ``exact error'', when the parameter \(\sigma \rightarrow 0.\)