A remark concerning divergence accuracy order for H(div)-conforming finite element flux approximations

TL;DR

This paper investigates the divergence accuracy order of H(div)-conforming finite element flux approximations, proposing a hierarchy of enriched schemes to improve divergence accuracy without increasing system complexity.

Contribution

It introduces a method to systematically enrich flux spaces for higher divergence accuracy in H(div) finite element methods, especially on non-affine geometries.

Findings

Enriched flux approximations achieve higher divergence accuracy orders.

The method maintains the same system size and structure as the original scheme.

Application to Darcy flow simulations demonstrates practical effectiveness.

Abstract

The construction of finite element approximations in $\mathbf{H}(\mbox{div}, {\Omega})$ usually requires the Piola transformation to map vector polynomials from a master element to vector fields in the elements of a partition of the region {\Omega}. It is known that degradation may occur in convergence order if non affine geometric mappings are used. On this point, we revisit a general procedure for the improvement of two-dimensional flux approximations discussed in a recent paper of this journal (Comput. Math. Appl. 74 (2017) 3283-3295). The starting point is an approximation scheme, which is known to provide $L^2$-errors with accuracy of order $k+1$ for sufficiently smooth flux functions, and of order $r+1$ for flux divergence. An example is $RT_{k}$ spaces on quadrilateral meshes, where $r = k$ or $k-1$ if linear or bilinear geometric isomorphisms are applied. Furthermore, the original space is required to be expressed by a factorization in terms of edge and internal shape flux functions. The goal is to define a hierarchy of enriched flux approximations to reach arbitrary higher orders of divergence accuracy $r+n+1$ as desired, for any $n \geq 1$. The enriched versions are defined by adding higher degree internal shape functions of the original family of spaces at level $k+n$, while keeping the original border fluxes at level $k$. The case $n=1$ has been discussed in the mentioned publication for two particular examples. General stronger enrichment $n>1$ shall be analyzed and applied to Darcy's flow simulations, the global condensed systems to be solved having same dimension and structure of the original scheme.