Lowest order methods for diffusive problems on general meshes: A unified approach to definition and implementation
In this work we propose an original point of view on lowest order methods for diffusive problems which lays the pillars of a C + + multi-physics, FreeFEM-like platform. The key idea is to regard lowest order methods as (Petrov)-Galerkin methods based on possibly incomplete, broken polynomial spaces defined from a gradient reconstruction. After presenting some examples of methods entering the framework, we show how implementation strategies common in the finite element context can be extended relying on the above definition. Several examples are provided throughout the presentation, and programming details are often omitted to help the reader unfamiliar with advanced C + + programming techniques.
KeywordsLowest-order methods Domain specific embedded language Petrov-Galerkin methods cell centered Galerkin methods hybrid finite volume methods
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Fruitful discussions with Christophe Prud’homme (Laboratoire Jean Kuntzmann, University of Grenoble) are gratefully acknowledged. We also wish to thank all the contributors to the Arcane platform.
- 3.L. Agélas, D. A. Di Pietro, and J. Droniou. The G method for heterogeneous anisotropic diffusion on general meshes. M2AN Math. Model. Numer. Anal., 44(4):597–625, 2010.Google Scholar
- 6.F. Brezzi, K. Lipnikov, and V. Simoncini. A family of mimetic finite difference methods on polygonal and polyhedral meshes. M3AS, 15:1533–1553, 2005.Google Scholar
- 7.I. Danaila, F. Hecht, and O. Pironneau. Simulation numérique en C++. Dunod, Paris, 2003. http://www.freefem.org.
- 8.D. A. Di Pietro. Cell centered Galerkin methods. C. R. Acad. Sci. Paris, Ser. I, 348:31–34, 2010.Google Scholar
- 9.D. A. Di Pietro. Cell centered Galerkin methods for diffusive problems. Submitted. Preprint available at http://hal.archives-ouvertes.fr/hal-00511125/en/, September 2010.
- 10.D. A. Di Pietro. A compact cell-centered Galerkin method with subgrid stabilization. C. R. Acad. Sci. Paris, Ser. I., 348(1–2):93–98, 2011.Google Scholar
- 11.D. A. Di Pietro and A. Ern. Mathematical aspects of discontinuous Galerkin methods. Mathematics & Applications. Springer-Verlag, Berlin, 2010. To appear.Google Scholar
- 14.J. Droniou, R. Eymard, T. Gallouët, and R. Herbin. A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. M3AS, Math. Models Methods Appl. Sci., 20(2):265–295, 2010.Google Scholar
- 18.G. Grospellier and B. Lelandais. The Arcane development framework. In Proceedings of the 8th workshop on Parallel/High-Performance Object-Oriented Scientific Computing, pages 4:1–4:11, New York, NY, USA, 2009. ACM.Google Scholar
- 19.A. Logg and G. N. Wells. DOLFIN: Automated finite element computing. ACM TOMS, 37, 2010.Google Scholar
- 20.C. Prud’homme. A domain specific embedded language in C++ for automatic differentiation, projection, integration and variational formulations. Sci. Prog., 14(2):81–110, 2006.Google Scholar