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 eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices arising from the p-version of finite element analysis. Bounds are derived for the eigenvalues and the condition numbers, which are valid for stiffness matrices based on a set of general basis functions that can be used in the p-version. For a set of hierarchical basis functions satisfying the usual local support condition that has been popularly used in the p-version, explicit bounds are derived for the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices. We prove that the condition numbers of the stiffness matrices grow like p4(d−1), where d is the number of dimensions. Our results disprove a conjecture of Olsen and Douglas in which the authors assert that “regardless of the choice of basis, the condition numbers grow like p4d or faster”. Numerical results are also presented which verify that our theoretical bounds are correct.