Raviart-Thomas finite elements of Petrov-Galerkin type

TL;DR

This paper introduces a Petrov-Galerkin variant for Raviart-Thomas finite elements in the Poisson problem, ensuring local gradient computation and establishing equivalence with finite volume and mass lumping schemes.

Contribution

It proposes a new Petrov-Galerkin approach with dual test functions for Raviart-Thomas elements, ensuring stability and convergence, and linking to existing finite volume methods.

Findings

The scheme is equivalent to the four point finite volume method.

Constructed dual test functions satisfy stability conditions.

Proved convergence using mixed finite element techniques.

Abstract

The mixed finite element method for the Poisson problem with the Raviart-Thomas elements of low-level can be interpreted as a finite volume method with a non-local gradient. In this contribution, we propose a variant of Petrov-Galerkin type for this problem to ensure a local computation of the gradient at the interfaces of the elements. The shape functions are the Raviart-Thomas finite elements. Our goal is to define test functions that are in duality with these shape functions: Precisely, the shape and test functions will be asked to satisfy a L2-orthogonality property. The general theory of Babu\v{s}ka brings necessary and sufficient stability conditions for a Petrov-Galerkin mixed problem to be convergent. We propose specific constraints for the dual test functions in order to ensure stability. With this choice, we prove that the mixed Petrov-Galerkin scheme is identical to the four point finite volumes scheme of Herbin, and to the mass lumping approach developed by Baranger, Maitre and Oudin. Finally, we construct a family of dual test functions that satisfy the stability conditions. Convergence is proven with the usual techniques of mixed finite elements.