Exponentially-fitted finite elements for $H({\rm curl})$ and $H({\rm div})$ convection-diffusion problems

TL;DR

This paper introduces exponentially-fitted finite element spaces for $H(curl)$ and $H(div)$ convection-diffusion problems on 3D meshes, enabling stable discretizations with flux approximation and complex structure preservation.

Contribution

It develops a novel construction of lowest order exponentially-fitted finite element spaces for $H(curl)$ and $H(div)$, including fluxes, and establishes a discrete convection-diffusion complex with proven properties.

Findings

Successfully constructs finite element spaces with fluxes on 3D meshes.

Establishes a discrete convection-diffusion complex with exactness.

Demonstrates framework commutativity under constant convection fields.

Abstract

This paper presents a novel approach to the construction of the lowest order $H(\mathrm{curl})$ and $H(\mathrm{div})$ exponentially-fitted finite element spaces ${\mathcal{S}_{1^-}^{k}}~(k=1,2)$ on 3D simplicial mesh for corresponding convection-diffusion problems. It is noteworthy that this method not only facilitates the construction of the functions themselves but also provides corresponding discrete fluxes simultaneously. Utilizing this approach, we successfully establish a discrete convection-diffusion complex and employ a specialized weighted interpolation to establish a bridge between the continuous complex and the discrete complex, resulting in a coherent framework. Furthermore, we demonstrate the commutativity of the framework when the convection field is locally constant, along with the exactness of the discrete convection-diffusion complex. Consequently, these types of spaces can be directly employed to devise the corresponding discrete scheme through a Petrov-Galerkin method.