Abstract
In this paper, we explore a method for the construction of locally conservative flux fields. The flux values are obtained through the use of a Ritz formulation in which we augment the resulting linear system of the continuous Galerkin (CG) formulation in a higher-order approximation space. These methodologies have been successfully applied to multi-phase flow models with heterogeneous permeability coefficients that have high-variation and discontinuities. The increase in accuracy associated with the high order approximation of the pressure solutions is inherited by the flux fields and saturation solutions. Our formulation allows us to use the saddle point problems analysis to study approximation and stability properties as well as iterative methods design for the resulting linear system. In particular, here we show that the low-order finite element problem preconditions well the high-order conservative discrete system. We present numerical evidence to support our findings.
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The use of second order formulation makes sense especially for cases where some form of high regularity holds. Usually in these cases the equality in the second order formulation is an equality in L 2 so that, in principle, there is no need to write the system of first order equations and weaken the equality by introducing less regular spaces for the pressure as it is done in mixed formulation with L 2 pressure.
References
E. Abreu, C. Díaz, J. Galvis, M. Sarkis, On high-order conservative finite element methods. Comput. Math. Appl. (Online December 2017). https://doi.org/10.1016/j.camwa.2017.10.020
M. Benzi, G.H. Golub, J. Liesen, Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)
L. Chen, A new class of high order finite volume methods for second order elliptic equations. SIAM J. Numer. Anal. 47(6), 4021–4043 (2010)
Z. Chen, J. Wu, Y. Xu, Higher-order finite volume methods for elliptic boundary value problems. Adv. Comput. Math. 37(2), 191–253 (2012)
Z. Chen, Y. Xu, Y. Zhang, A construction of higher-order finite volume methods. Math. Comput. 84, 599–628 (2015)
M. Presho, J. Galvis, A mass conservative generalized multiscale finite element method applied to two-phase flow in heterogeneous porous media. J. Comput. Appl. Math. 296, 376–388 (2016)
Acknowledgements
E. Abreu and C. Díaz thank for the financial support by FAPESP through grant No. 2016/23374-1. M. Sarkis is supported by NSF DMS-1522663.
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Galvis, J., Abreu, E., Díaz, C., Sarkis, M. (2018). On High-Order Approximation and Stability with Conservative Properties. In: Bjørstad, P., et al. Domain Decomposition Methods in Science and Engineering XXIV . DD 2017. Lecture Notes in Computational Science and Engineering, vol 125. Springer, Cham. https://doi.org/10.1007/978-3-319-93873-8_23
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DOI: https://doi.org/10.1007/978-3-319-93873-8_23
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