An irregular grid for the numerical solution of linear elliptic partial differential equations (Q2485610)

The authors show that the condition number of the symmetrically scaled stiffness matrix \(\overline A= DAD^T\) behaves like \(O(1)\), where \(D= (\text{diag}(A))^{-1/2}\) and the stiffness matrix \(A\) is obtained from the three-point centered difference approximation of the homogeneous Dirichlet two-point boundary value problem \(-u''= f\) in \((0, 1)\) on a boundary concentrated grid with a fixed grid grading parameter \(1/\alpha\in(0, 1)\) towards the two boundary points \(x= 0\) and \(x= 1\). This theoretical result is only proved for the 1D case, whereas the numerical results show a similar behavior for the 2D analog, too. The reviewer wants to add that \textit{H. Yserentant} [Numer. Algorithms 21, No. 1--4, 387--392 (1999; Zbl 0939.65134)], \textit{B. N. Khoromskij} and \textit{J. M. Melenk} [SIAM J. Numer. Anal. 41, No. 1, 1--36 (2003; Zbl 1050.65113)] and others have introduced and investigated boundary concentrated finite element methods in earlier papers which are not mentioned in this paper.