Mesh selection for a nearly singular boundary value problem (Q5960024)

The authors investigate the numerical solution of the model equation \(u_{xx}=\varepsilon^{-2}e^{-x/\varepsilon }\) (\(x\in (0,1)\)) with \(u(0)=1\) and \(u(1)=\alpha =e^{-1/\varepsilon }\) and several slightly more general problems when \(\varepsilon << 1\) using the standard central difference scheme on nonuniform grids. In particular, it is shown the error behavior in two limiting cases: (i) the total mesh point number \(N\) is fixed when the regularization parameter \(\varepsilon \to 0\); (ii) \(\varepsilon \) is fixed when \(N\to \infty \). Using a formal analysis, the authors show that a generalized version of a special piecewise uniform mesh and an adaptive grid based on the equidistribution principle share some common features. The optimal meshes give rates of convergence bounded by \(|\log (\varepsilon)|\) as \(\varepsilon\to 0\) and \(N\) is given, which are shown to be sharp by numerical tests.