Uniform convergence for the difference scheme in the conservation form of ordinary differential equation with a small parameter (Q1091779)

For the singular perturbation problem of the selfadjoint ordinary differential equation \[ (1)\quad Lu(x)=-\epsilon (p(x)u'(x))'+q(x)u(x)=f(x)\quad (x\in 0,1)),\quad u(0)=A,\quad u(1)=B \] with \(p(x)\geq \alpha >0\), \(q(x)\geq \beta >0\) (x\(\in [0,1])\) and \(\epsilon >0\) being a small parameter a class of difference schemes in conservation form with exponentially fitted factors \(\sigma_ i(h/\sqrt{\epsilon})\) is constructed. Sufficient conditions for \(\sigma_ i\) are derived to satisfy the purpose that the corresponding scheme is uniformly convergent to the solution of (1). In the case at issue the order of convergence is one, and under some assumption on p(x) and q(x), it is two.