Effectively well-conditioned linear systems
SIAM Journal on Scientific and Statistical Computing
The algebraic eigenvalue problem
The algebraic eigenvalue problem
On the conditioning of finite element equations with highly refined meshes
SIAM Journal on Numerical Analysis
Mathematical Software
Error analysis of the Trefftz method for solving Laplace's eigenvalue problems
Journal of Computational and Applied Mathematics
Effective condition number for finite difference method
Journal of Computational and Applied Mathematics
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This is a continued study but at advanced levels of effective condition number in [Z.C. Li, C.S. Chien, H.T. Huang, Effective condition number for finite difference method, J. Comput. Appl. Math. 198 (2007) 208-235; Z.C. Li, H.T. Huang, Effective condition number for numerical partial differential equations, Numer. Linear Algebra Appl. 15 (2008) 575-594] for stability analysis. To approximate Poisson's equation with singularity by the finite element method (FEM), the adaptive mesh refinements are an important and popular technique, by which, the FEM solutions with optimal convergence rates can be obtained. The local mesh refinements are essential to FEM for solving complicated problems with singularities, and they have been used for three decades. However, the traditional condition number is given by Cond=O(h"m"i"n^-^2) in Strang and Fix [G. Strang, G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973], where h"m"i"n is the minimal length of elements. Since h"m"i"n is infinitesimal near the singular points, Cond is huge. Such a dilemma can be bypassed by small effective condition number. In this paper, the bounds of the simplified effective condition number Cond_EE are derived as O(1), O(h^-^1^.^5) or O(h^-^0^.^5), where h(