An analysis of some element-by-element techniques
The paper provides an analysis of some EBE preconditioners \(P_{EBE}\) originally introduced by \textit{T. J. R. Hughes}, \textit{I. Levit} and \textit{J. Winger} [ibid. 36, 241-354 (1983; Zbl 0487.73083)]. The EBE preconditioning effects are compared with those obtained by diagonal scaling. This comparison is made for finite element approximations \(Ax=b\) to the simple model problem \(-\Delta u=f\) in \(\Omega =(0,1)^ d\) and \(u=0\) on \(\partial \Omega\) for \(d=1,2,3\). On the basis of the numerical computation of the corresponding spectral condition numbers \(\kappa\) (.) for finer and finer grids, the author observes that \(\kappa (P^{- 1}_{diag}A)\simeq \alpha \kappa (P^{-1}_{EBE}A)\) with some \(\alpha\) between 3 and 9, where \(P_{diag}=diag(A).\) This relation would imply that the EBE preconditioners \(P_{EBE}\) cannot improve the asymptotic behaviour of \(\kappa (P^{-1}_{EBE}A)\) in comparison with \(\kappa\) (A). For instance, for second-order elliptic boundary value problems, we would have \(\kappa (P^{- 1}_{EBE}A)=O(\kappa (A))=O(h^{-2})\) if the discretization parameter h \((h=O(n^{-1/d}_{nodes}))\) tends to zero. Thus, it would be desirable to analyze the EBE ``preconditioners'' for more complicated problems (strongly discontinuous coefficients, irregular domains, systems of PDEs etc.) instead of the model problems considered in this paper.
scientific article; zbMATH DE number 4137241
scientific article