Raviart Thomas Petrov-Galerkin Finite Elements
This work provides a theoretical unification of several numerical methods for solving mixed problems, but the contribution is primarily theoretical and incremental, as the resulting scheme is equivalent to existing methods.
The authors propose a Petrov-Galerkin variant of Raviart-Thomas mixed finite elements to achieve local gradient computation at element interfaces. They prove that the resulting method is identical to existing finite volume and mass condensation schemes, and demonstrate stability via an inf-sup condition and convergence using standard mixed finite element techniques.
The general theory of Babuška 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 à 4 points" ("VF4") of Faille, Galloüet 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.