Sparsity optimized high order finite element functions for H(div) on simplices

Abstract

This paper deals with conforming high-order finite element discretizations of the vector-valued function space H(Div) in 2 and 3 dimensions. A new set of hierarchic basis functions on simplices with the following two main properties is introduced. When working with an affine, simplicial triangulation, first, the divergence of the basis functions is L 2-orthogonal, and secondly, the L 2-inner product of the interior basis functions is sparse with respect to the polynomial degree p. The construction relies on a tensor-product based construction with properly weighted Jacobi polynomials as well as an explicit splitting of the higher-order basis functions into solenoidal and non-solenoidal ones. The basis is suited for fast assembling strategies. The general proof of the sparsity result is done by the assistance of computer algebra software tools. Several numerical experiments document the proved sparsity patterns and practically achieved condition numbers for the parameter-dependent Div - Div problem. Even on curved elements a tremendous improvement in condition numbers is observed. The precomputed mass and stiffness matrix entries in general form are available online.