Effective condition number of the Hermite finite element methods for biharmonic equations

For biharmonic equations, the Hermite finite element methods (FEM) are chosen, to seek their approximate solutions. The linear algebraic equations Ax=b are obtained from the Hermite FEM, where the matrix A is symmetric and positive definite, and x and b are the unknown and known vectors, respectively. It is well known that , and λmax and λmin are the maximal and minimal eigenvalues of the stiffness matrix A, respectively. The bounds of Cond are derived to be O(h−4). Note that when h is small, the values of Cond (=O(h−4)) are huge, to indicate a severe instability, compared with Cond =O(h−2) for Poisson's equation by the FEM. In fact, for specific application problems, the instability is not so severe, a new effective condition number is defined by Cond_eff in [Z.C. Li, C.S. Chien, H.T. Huang, Effective condition number for finite difference method, Comput. Appl. Math. 198 (2007) 208–235], to provide a better upper bound of perturbation errors. It is proven that Cond_eff =O(h−3.5) for general cases, which is smaller than the traditional Cond. However, for special cases, the Cond_eff could be much smaller. For instant, for the homogeneous boundary conditions of biharmonic equations, Cond_eff =O(1), can be reached as h diminishes. This is astonishing, against our intuition from the knowledge of the Cond. From the analysis in this paper, the traditional Cond may mislead the stability analysis for practical computation of engineering problems.