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Erschienen in:
2017 | OriginalPaper | Buchkapitel
Raviart Thomas Petrov–Galerkin Finite Elements
verfasst von : François Dubois, Isabelle Greff, Charles Pierre
Erschienen in: Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects
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Abstract
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.