On the Construction of Well-Conditioned Hierarchical Bases for Tetrahedral $\mathcal{H}{(curl)}$-Conforming Nédélec Elements

Abstract

A partially orthonormal basis is constructed with better conditioning properties for tetrahedral $\mathcal{H}{(curl)}$-conforming Nédélec elements. The shape functions are classified into several categories with respect to their topological entities on the reference 3-simplex. The basis functions in each category are constructed to achieve maximum orthogonality. The numerical study on the matrix conditioning shows that for the mass and quasi-stiffness matrices, and in a logarithmic scale the condition number grows linearly vs. order of approximation up to order three. For each order of approximation, the condition number of the quasi-stiffness matrix is about one order less than the corresponding one for the mass matrix. Also, up to order six of approximation the conditioning of the mass and quasi-stiffness matrices with the proposed basis is better than the corresponding one with the Ainsworth-Coyle basis Internat. J. Numer. Methods. Engrg., 58:2103-2130, 2003. except for order four with the quasi-stiffness matrix. Moreover, with the new basis the composite matrix $µM+S$ has better conditioning than the Ainsworth-Coyle basis for a wide range of the parameter $µ$.