High order biorthogonal functions in H(Curl)

TL;DR

This paper introduces biorthogonal basis functions for H(Curl) finite element methods, optimizing computational efficiency by leveraging tensor products of Jacobi polynomials on hypercubes and simplices.

Contribution

It develops biorthogonal basis functions compatible with existing primal bases, enhancing computational properties in hp-FEM for H(Curl) spaces.

Findings

Biorthogonal functions expressed as sums of tensor products of Jacobi polynomials.

Applicable to hypercubes and simplices as reference elements.

Improves condition number and sparsity in finite element computations.

Abstract

From the literature, it is known that the choice of basis functions in hp-FEM heavily influences the computational cost in order to obtain an approximate solution. Depending on the choice of the reference element, suitable tensor product like basis functions of Jacobi polynomials with different weights lead to optimal properties due to condition number and sparsity. This paper presents biorthogonal basis functions to the primal basis functions mentioned above. The authors investigate hypercubes and simplices as reference elements, as well as the cases of $H^1$ and H(Curl). The functions can be expressed sums of tensor products of Jacobi polynomials with maximal two summands.