A new procedure for the construction of hierarchical high order Hdiv and Hcurl finite element spaces

This paper considers a systematic procedure for the construction of a hierarchy of high order finite element approximation for Hdiv and Hcurl spaces based on triangular and quadrilateral partitions of bidimensional domains. The principle is to choose an appropriate set of vectors, based on the geometry of each element, which are multiplied by an available set of H1H1 hierarchical scalar basic functions. This strategy produces vector basis functions with continuous normal or tangent components on the elements interfaces, properties that characterise functions in Hdiv or Hcurl, respectively. We also present a numerical study to evaluate the correct balancedness of the resulting Hdiv spaces of degree kk and L2L2 spaces of degree k−1k−1 on the resolution of the mixed formulation for a Steklov eigenvalue problem.