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 $p^{4(d-1)}$, 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 $p^{4d}$ or faster". Numerical results are also presented which verify that our theoretical bounds are correct.