Fully hierarchical divergence‐conforming basis functions on tetrahedral cells, with applications

A new set of hierarchical, divergence-conforming, vector basis functions on curvilinear tetrahedrons is presented. The basis can model both mixed- and full-order polynomial spaces to arbitrary order, as defined by Raviart and Thomas, and Nedelec. Solenoidal- and non-solenoidal components are separately represented on the element, except in the case of the mixed first-order space, for which a decomposition procedure on the global, mesh-wide level is presented. Therefore, the hierarchical aspect of the basis can be made to extend down to zero polynomial order. The basis can be used to model divergence-conforming quantities, such as electromagnetic flux- and current density, fluid velocity, etc., within numerical methods such as the finite element method (FEM) or integral equation-based methods. The basis is ideally suited to p-adaptive analysis. The paper concludes with two example applications. The first is the FEM-based solution of the linearized acoustic vector wave equation, where it is shown how the decomposition into solenoidal components and their complements can be used to stabilize the method at low frequencies. The second is the solution of the electric field, volume integral equation for electromagnetic scattering analysis, where the benefits of the decomposition are again demonstrated.