Effectively well-conditioned linear systems
SIAM Journal on Scientific and Statistical Computing
The algebraic eigenvalue problem
The algebraic eigenvalue problem
The effective condition number applied to error analysis of certain boundary collocation methods
Journal of Computational and Applied Mathematics
The conditioning of some numerical methods for first kind boundary integral equations
Journal of Computational and Applied Mathematics
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Journal of Computational and Applied Mathematics
Boundary penalty finite element methods for blending surfaces — II: biharmonic equations
Journal of Computational and Applied Mathematics
Global superconvergence for blending surfaces by boundary penalty pluls hybrid FEMs
Applied Numerical Mathematics
New error estimates of bi-cubic Hermite finite element methods for biharmonic equations
Journal of Computational and Applied Mathematics
Effective condition number for finite difference method
Journal of Computational and Applied Mathematics
Finite element method for conserved phase fields: Stress-mediated diffusional phase transformation
Journal of Computational Physics
Mathematics and Computers in Simulation
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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 Cond=@l"m"a"x@l"m"i"n, and @l"m"a"x and @l"m"i"n 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 =@?b@?@?x@?@l"m"i"n 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.