The numerical solution of singular perturbations of boundary value problems.

*(English)*Zbl 0213.16704The author presents a comprehensive study of finite difference approximations to the boundary value problem
\[
u'' = f(t,u,v),\quad u(0)=u(1) =0,\quad \varepsilon v" + g(t,u,u')v' = 0,\quad v(0) = v_1, v(1) = v_1,
\]
with main emphasis on the behavior of the solutions as \(\varepsilon\to 0\). In a preliminary section difference approximations of positive type are introduced and a number of their properties are discussed. In particular, the validity of obtaining asymptotic solutions by letting first \(\varepsilon\to 0\) and then decreasing the mesh size is established. The author then concentrates on problems with interior turning points and proves that the finite difference solution reflects the asymptotic behavior of the continuous solution as \(\varepsilon\to 0\). The same result is shown to hold true for more general equations without turning points. The author also demonstrates that difference approximations of positive type are essential to obtain the correct asymptotic behavior. Next, the convergence of the finite difference solution to the classical solution for fixed \(\varepsilon\) and vanishing mesh size is proved. Finally, some comments on a numerical algorithm for nonlinear problems and some computer results are presented.

Reviewer: G. H. Meyer