Raviart Thomas Petrov-Galerkin Finite Elements

TL;DR

This paper introduces a Petrov-Galerkin finite element variant that ensures local gradient computation, demonstrating its equivalence to certain finite volume schemes and establishing stability and convergence.

Contribution

It proposes a new Petrov-Galerkin approach for Raviart-Thomas elements that guarantees local gradient computation and proves its stability and equivalence to existing finite volume methods.

Findings

The method is identical to the VF4 finite volume scheme.

The approach satisfies an inf-sup stability condition.

The scheme converges using standard mixed finite element techniques.

Abstract

The general theory of Babu\v{s}ka ensures necessary and sufficient conditions for a mixed problem in classical or Petrov-Galerkin form to be well posed in the sense of Hadamard. Moreover, the mixed method of Raviart-Thomas with low-level elements can be interpreted as a finite volume method with a non-local gradient. In this contribution, we propose a variant of type Petrov-Galerkin to ensure a local computation of the gradient at the interfaces of the elements. The in-depth study of stability leads to a specific choice of the test functions. With this choice, we show on the one hand that the mixed Petrov-Galerkin obtained is identical to the finite volumes scheme "volumes finis \`a 4 points" ("VF4") of Faille, Gallo\"uet and Herbin and to the condensation of mass approach developed by Baranger, Maitre and Oudin. On the other hand, we show the stability via an inf-sup condition and finally the convergence with the usual methods of mixed finite elements.