Variants of Raviart-Thomas mixed elements for curved domains using straight-edged tetrahedra

TL;DR

This paper extends Raviart-Thomas mixed finite element methods for curved domains from 2D to 3D, focusing on boundary flux enforcement and comparing different formulations through numerical experiments.

Contribution

It introduces a natural extension of a 2D boundary flux enforcement method to 3D tetrahedral meshes and evaluates its performance against existing strategies.

Findings

The 3D extension performs comparably to 2D methods.

Straight-edged tetrahedra with Petrov-Galerkin formulation improve boundary flux accuracy.

Hermite analog shows enhanced performance in lowest order methods.

Abstract

A numerical study of tetrahedral Raviart-Thomas mixed finite element methods is presented in the solution of model second order boundary value problems posed in a curved spatial domain. An emphasis is given to the case where normal fluxes are prescribed on a boundary portion. In this case the question on the best way to enforce known boundary degrees of freedom is raised. It seems intuitive that the normal component of the flux variable should preferably not take up corresponding prescribed values at nodes shifted to the boundary of the approximating polyhedron in the underlying normal direction. This is because an accuracy downgrade is to be expected, as shown in https://doi.org/10.1137/15M1045442 and https://doi.org/10.1051/m2an/2025028. In the former work accuracy improvement is achieved by means of a standard Galerkin formulation with parametric elements. The latter one in turn advocates the use of straight-edged triangles combined with a Petrov-Galerkin formulation, in which the aforementioned shift applies only to the test-flux space, while the shape-flux space consists of fields whose fluxes satisfy the prescribed conditions on the true boundary. The first purpose of this article is to show that the method studied in https://doi.org/10.1051/m2an/2025028 for two-dimensional problems can be extended quite naturally to the three-dimensional case. More particularly we illustrate this by carrying out numerical experimentation with such a version for the two lowest order methods of this family, as compared to the corresponding do-nothing strategy. In the case of the lowest order method this comparative study is enriched by assessing as well the performance of its Hermite analog introduced in https://doi.org/10.1016/j.cam.2012.08.027.