Solving singularly perturbed boundary-value problems by spline in tension (Q1115125)

The authors present a difference scheme on the basis of the spline in tension for the boundary value problem \(-\epsilon y''+p(x)y=f(x),\) where \(p(x)>0\), \(y(0)=\alpha\), \(y(1)=\beta\). It is proved that the error is of the form \(O(h\times \min (h,\sqrt\epsilon))\) for arbitrary p(x), and in the case of \(p(x)=const.\) the corrsponding spline in tension achieves second order of global uniform convergence, i.e. the error is of the form \(O(h^ 2)\). A numerical example confirms the theoretical results.