BOUNDS FOR EIGENVALUES AND CONDITION NUMBERS IN THE P-VERSION OF THE FINITE-ELEMENT METHOD

In this paper, we present a theory for bounding the minimum eigenvalue s, maximum eigenvalues, and condition numbers of stiffness matrices ar ising from the p-version of finite element analysis. Bounds are derive d for the eigenvalues and the condition numbers, which are valid for s tiffness matrices based on a set of general basis functions that can b e used in the p-version. For a set of hierarchical basis functions sat isfying the usual local support condition that has been popularly used in the p-version, explicit bounds are derived for the minimum eigenva lues, maximum eigenvalues, and condition numbers of stiffness matrices . We prove that the condition numbers of the stiffness matrices grow l ike p(4(d-1)), where d is the number of dimensions. Our results dispro ve a conjecture of Olsen and Douglas in which the authors assert that ''regardless of the choice of basis, the condition numbers grow like p (4d) or faster''. Numerical results are also presented which verify th at our theoretical bounds are correct.