Multigrid applied to singular perturbation problems
The solution of the singular perturbation problem (1) \(-\epsilon u''+b(x)u'=f\), \(u(0)=u_ 0\), \(u(1)=u_ 1\), \(1>>\epsilon >0\) via a multigrid algorithm is considered. Three model problems are studied: BL(b(x) does not change sign, problem BL exhibits a boundary layer at \(x=1)\), TP-1, TP-2 (b(x) changes sign, there are two boundary layers at \(x=0\), and \(x=1\), problems with turning point at \(x=0.5)\). A cursory review from the previous paper by the author and \textit{S. V. Parter} [ibid. 17, 153-184 (1985; Zbl 0608.65050)] on a multigrid algorithm used to solve (1) is given. Some results for convergence conclusions are repeated. Theoretical and experimental results for a number of different discretizations are presented. Many practical (from a computational point of view) but also theoretical questions are answered in comments on experimental results for BL, TP-1 and TP-2 problems.
scientific article; zbMATH DE number 4043177