Mesh Selection for a Nearly Singular Boundary Value Problem

AbstractIn this paper, we investigate the numerical solution of a model equation uxx= $$\frac{1}{{\varepsilon ^2 }}$$ exp(− $$\frac{x}{\varepsilon }$$ ) (and several slightly more general problems) when ∈≪1 using the standard central difference scheme on nonuniform grids. In particular, we are interested in the error behaviour in two limiting cases: (i) the total mesh point number N is fixed when the regularization parameter ∈→0, and (ii) ∈ is fixed when N→∞. Using a formal analysis, we show that a generalized version of a special piecewise uniform mesh 12 and an adaptive grid based on the equidistribution principle share some common features. And the “optimal” meshes give rates of convergence bounded by |log(∈)| as ∈→0 and N is given, which are shown to be sharp by numerical tests.