Raviart-Thomas finite elements of Petrov-Galerkin type
For researchers in numerical methods for PDEs, this work provides a theoretical unification of mixed finite elements and finite volume methods, but the result is incremental as it reproduces known schemes.
The paper proposes a Petrov-Galerkin variant of Raviart-Thomas finite elements for the Poisson problem, achieving local gradient computation at element interfaces. The method is shown to be equivalent to existing finite volume and mass lumping schemes, with proven convergence.
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 Babuška 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.