An Analysis of a Uniformly Accurate Difference Method for a Singular Perturbation Problem

It will be proven that an exponential tridiagonal difference scheme, when applied with a uniform mesh of size h to: $\varepsilon {u_{xx}} + b(x){u_x} = f(x)$ for $0 < x < 1,b > 0$, b and f smooth, $\varepsilon$ in (0, 1], and $u(0)$ and $u(1)$ given, is uniformly second-order accurate (i.e., the maximum of the errors at the grid points is bounded by $C{h^2}$ with the constant C independent of h and $\varepsilon$). This scheme was derived by El-Mistikawy and Werle by a ${C^1}$ patching of a pair of piecewise constant coefficient approximate differential equations across a common grid point. The behavior of the approximate solution in between the grid points will be analyzed, and some numerical results will also be given.