On High-Order Approximation and Stability with Conservative Properties
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.
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.
- 1.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