Efficient implementation of the hybridized Raviart-Thomas mixed method by converting flux subspaces into stabilizations
Abstract: We show how to reduce the computational time of the practical implementation of the Raviart-Thomas mixed method for second-order elliptic problems. The implementation takes advantage of a recent result which states that certain local subspaces of the vector unknown can be eliminated from the equations by transforming them into stabilization functions; see the paper published online in JJIAM on August 10, 2023. We describe in detail the new implementation (in MATLAB and a laptop with Intel(R) Core (TM) i7-8700 processor which has six cores and hyperthreading) and present numerical results showing 10 to 20% reduction in the computational time for the Raviart-Thomas method of index $k$, with $k$ ranging from 1 to 20, applied to a model problem.
- D.N. Arnold and F. Brezzi. Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. Modél. Math. Anal. Numér., 19:7–32, 1985.
- Algorithm 887: Cholmod, supernodal sparse Cholesky factorization and update/downdate. ACM Transactions on Mathematical Software (TOMS), 35(3):1–14, 2008.
- B. Cockburn. Static condensation, hybridization, and the devising of the HDG methods. In G.R. Barrenechea, F. Brezzi, A. Cagniani, and E.H. Georgoulis, editors, Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, volume 114 of Lect. Notes Comput. Sci. Engrg., pages 129–177. Springer Verlag, Berlin, 2016. LMS Durham Symposia funded by the London Mathematical Society. Durham, U.K., on July 8–16, 2014.
- B. Cockburn. Discontinuous Galerkin methods for computational fluid dynamics. In E. Stein, R.de Borst, and T.J.R. Hughes, editors, Encyclopedia of Computational Mechanics Second Edition, volume 5, pages 141–203. John Wiley & Sons, Ltd., Chichester, U.K., 2018.
- B. Cockburn. Hybridizable discontinuous Galerkin methods for second-order elliptic problems: Overview, a new result and open problems. Japan J. Indust. Appl. Math., published online on August 10th, 2023. https//doi.org/10.1007/s13160-023-00603-9.
- An introduction to the theory of M𝑀Mitalic_M-decompositions. In Numerical methods for PDEs, volume 15 of SEMA SIMAI Springer Ser., pages 5–29. Springer, Cham, 2018.
- Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal., 47:1319–1365, 2009.
- M. Dubiner. Spectral methods on triangles and other domains. Journal of Scientific Computing, 6(4):345–390, 1991.
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