Hierarchical polynomials and vector elements for finite methods

This paper presents a new set of hierarchical vector elements of arbitrarily high polynomial order constructed by using new orthogonal scalar polynomials. These novel vector elements, with respect to existing ones, provide better conditioned system matrices in finite methods applications. The scalar polynomials are subdivided into edge-, face-, and volume-based polynomials. In each group, all the polynomials are mutually orthogonal independent of the definition domain of the inner product, i.e. either the volume, the face, or the edge of the element. The good properties of these new vector elements are confirmed by numerical results.