Bounds on spectral condition numbers of matrices arising in the $p$-version of the finite element method

Summary.

We estimate condition numbers of $p$-version matrices for tensor product elements with two choices of reference element degrees of freedom. In one case (Lagrange elements) the condition numbers grow exponentially in $p$, whereas in the other (hierarchical basis functions based on Tchebycheff polynomials) the condition numbers grow rapidly but only algebraically in $p$. We conjecture that regardless of the choice of basis the condition numbers grow like $p^{4d}$ or faster, where $d$ is the dimension of the spatial domain.