Improved conditioning of finite element matrices using new high-order interpolatory bases

The condition number of finite element matrices constructed from interpolatory bases will grow as the polynomial degree of the basis functions is increased. The worst case scenario for this growth rate is exponential and in this paper we demonstrate through computational example that the traditional set of uniformly distributed interpolation points yields this behavior. We propose a set of nonuniform interpolation points which yield a much improved polynomial growth rate of condition number. These points can be used to construct several types of popular hexahedral basis functions including the 0-form (standard Lagrangian), 1-form (Curl conforming), and 2-form (Divergence conforming) varieties. We demonstrate through computational example the benefits of using these new interpolatory bases in finite element solutions to Maxwell´s equations in both the frequency and time domain.