Strong uniform stability and exact discretizations of a model singular perturbation problem and its finite-difference approximations
The author studies both the strong uniform stability and the norm \(\|| u\||_{\epsilon}:=\| u\|_{\infty}+\delta \| u'\|_{\infty}\) and three-point approximation in the norm \(\|| u^ H\||_{H,\epsilon}:=\| u^ H\|_{H,\infty}+\epsilon \| Du^ H\|_{H,\infty}\) for the model problem \[ (1)\quad L_{\epsilon}u:=-\epsilon u''+a(x)u'+b(x)u=f(x),\quad 0<x<1,\quad u(0)=A,\quad u(1)=B, \] where \(\epsilon >0\), \(a\in L^{\infty}[0,1]\), \(b,f\in L^ 1[0,1]\), \(a(x)\geq a>0\). The strong uniform stability of the discrete schemes is analyzed using exact three-point schemes for (1). It should be pointed out that the exact three-point difference schemes for differential equations of second order were introduced by \textit{A. N. Tikhonov} and \textit{A. A. Samarskii} [Zh. Vychisl. Mat. Mat. Fiz. 1, 425- 440 (1961; Zbl 0125.073)]. Three-point difference approximation of (1), which have been build on this ground and possess uniform speed of convergence were given by \textit{M. V. Alekseevskij} [Difference methods in mathematical physics, Collect. Pap., Moscow 1979, 36-60 (1979; Zbl 0456.34022)].
scientific article; zbMATH DE number 4115868
scientific article