Numerical solution of the singularly perturbed problem with nonlocal boundary condition (Q1861035)

This paper is concerned with the numerical solution by a finite difference method of the following singularly perturbed nonlocal boundary value problem: \[ -\varepsilon u'' + a(x) u = f(x),\quad 0<x<l, \] \[ -\sqrt{\varepsilon} u'(0) + \gamma u(0) = \mu_0, \] \[ u(l) - \delta u(d) = \mu_l, \] where \(d\) is some given point in \((0,l)\). Moreover, \(\varepsilon\) is a small positive parameter, \(\gamma>0\), \(\delta>0\), \(\mu_0\) and \(\mu_l\) are given constants, and the functions \(a(x)\geq \alpha >0\) and \(f(x)\) are smooth enough so that the problem has a unique solution \(u(x)\) for all small values of \(\varepsilon\). This solution has in general boundary layers at \(x=0\) and \(x=l\) for \(\varepsilon\) near \(0\), which causes trouble when classical numerical methods are applied. In this note the authors present a uniformly convergent difference scheme on a non-equidistant mesh, based on the method of integral identities, by using a basis of exponential functions and interpolating quadrature rules with the weight and the remainder terms in integral form. Uniform error estimates are established in the discrete maximum norm, and a numerical example is given.