Condition number and diagonal preconditioning: comparison of the $p$-version and the spectral element methods

Summary.In the framework of adaptive methods, bases of hierarchical type are used in the $p$-version of the finite element method. We study the matrices corresponding to the commonly used basis, introduced by Babuška and Szabo, in the case of $d$-dimensional rectangular elements for 2 $^{\rm nd}$ order elliptic boundary value problems. For the internal nodes, we show that the condition number is equivalent to $p^{4 (d-1)}$ and to $p^{4d}$ for the stiffness and mass matrix, respectively. Moreover, we show that the usual diagonal preconditioning divides in the previous orders the exponents of $p$ by two. Finally, we compare these results with those obtained for spectral elements (nodal basis).