Effective condition number for the finite element method using local mesh refinements

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 in Strang and Fix [G. Strang, G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973], where hmin is the minimal length of elements. Since hmin 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 is the maximal length of elements. Evidently, Cond_EE is much smaller than Cond. The numerical experiments are carried out, to verify the stability analysis. Small effective condition numbers explain well the satisfactory FEM solutions obtained. This paper provides a stability justification for the adaptive mesh refinements used in FEM. Compared with [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], the analysis in this paper is more difficult and challenging, its proof techniques are new and intriguing, and the results are more important and useful.