THE l2 AND l∞ CONDITION NUMBERS OF THE FINITE ELEMENT STIFFNESS AND MASS MATRICES, AND THE POINTWISE CONVERGENCE OF THE METHOD

This chapter discusses the l 2 and l ∞ condition numbers of the finite element stiffness and mass matrices, and the pointwise convergence of the method. Computable bounds are derived for the extremal eigenvalues of the global mass and stiffness matrices generated by the method of finite elements. With these bounds, it becomes possible to bound the spectral ( l 2 ) condition numbers C 2 ( K ) = ∥ K ∥ 2 ∥ K –1 ∥ 2 of the stiffness matrix K , and C 2 ( M ) = ∥ μ ∥ 2 ∥ μ –1 ∥ 2 of the mass (Gram) matrix M . The bounds on the condition numbers are expressed in terms of the extremal eigenvalues of the element stiffness and mass matrices, the maximum number of elements meeting at a nodal point and the fundamental frequency of the structure. As the element matrices are of restricted dimension, the dependence of their eigenvalues on the geometry of the element can be readily established either numerically or even algebraically. These bounds on C 2 ( K ) and C 2 ( M ) are next applied to second and fourth order problems, and to some inherently ill-conditioned problems, to determine the influence of a problem's intrinsic and discretization parameters on the condition of K and M .