A priori meshes for singularly perturbed quasilinear two-point boundary value problems (Q2713132)

The problem of a proper choice of discretisation meshes for the solution of the differential equation having boundary layers is presented on the simple model of a singularly perturbed ordinary differential equation of the form NEWLINE\[NEWLINE\varepsilon u''-b(x,u)u'+c(x,u)=0,\;\;x\in [0,1],\;\;0<\varepsilon\ll 1, \tag \(*\) NEWLINE\]NEWLINE with boundary conditions NEWLINE\[NEWLINEu(0)=u(1)=0.NEWLINE\]NEWLINE Let \(u_{\varepsilon}\) be a solution of the singularly perturbed problem \((*)\). In order to recognize character and position of the layers an estimation of the difference \(|u_{\varepsilon}-u_0|\) in terms of the variable \(x\) and the singular parameter \(\varepsilon\) is used. For example in the discussed case, under certain additional assumptions, this estimate takes the form NEWLINE\[NEWLINE|u_{\varepsilon}-u_0|\leq M(\varepsilon+e^{-\beta {x\over \varepsilon}}).NEWLINE\]NEWLINE This means that the boundary layer is situated near \(0\) and that it is of the exponential character. If positions and characters of the layers are known we may think about an a priori choice of discretisation meshes. Two general approches are discussed: the \(B\)-meshes by \textit{N. S. Bakhalov} [Zh. Vychisl. Mat. Mat. Fiz. 9, 841-859 (1969; Zbl 0208.19103)] and proposed by the author's generalization of \(S(l)\)-meshes originaly introduced by \textit{G. I. Shishkin} [ibid. 26, No. 11, 1649-1662 (1988; Zbl 0662.65086)]. The \(B\)-meshes are generated by some given continuous function and hence are essentially non-uniform, while the \(S(l)\)-meshes are by definition piecewise uniform. The advantage of \(B\)-meshes is a good adaptation to the character of the layer and hence a good convergence of the discretised solution to the original one. However the non-uniform discretisation has well known bad sides. Piecewise uniform discretization is somewhat more convenient, but using it one has to pay by worse convergence. This is evident for the original Shishkin construction. The new concept by the author attenuates this bad effect. Other aspects of the invoked problems are also discussed. The paper contains numerical examples.