Bounds for eigenvalues and condition numbers in the 𝑝-version of the finite element method

Abstract

In this paper, we present a theory for bounding the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices arising from the p 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 p -version. For a set of hierarchical basis functions satisfying the usual local support condition that has been popularly used in the p 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 ) p^{4(d-1)} , where d 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 4 d p^{4d} or faster". Numerical results are also presented which verify that our theoretical bounds are correct.