Journal "Computational Technologies"

Article information

2018 , Volume 23, № 1, p.46-60

Itkina N.B., Trofimova S.A.

Application of mixed non-conforming variational formulations for solving the Darcy problem

We consider the construction of a computational scheme for the Darcy model with a tensor permeability coefficient. We use the mixed finite element approximation based on the discontinuous Galerkin formulation. Recently the mixed method has become one of the modern approaches for a numerical solution of the Darcy problem. The main idea of this method is to approximate both the primary and dual variables, while searching for the critical point of the corresponding functional over the finite-element space of admissible test functions, which can be represented as a direct sum of two or more subspaces. This approach permits us to find a solution that corresponds to the physics of the simulated processes. The advantages and disadvantages of the mixed method are discussed in this paper.

We use numerical fluxes and jump stabilization following the approach of F. Brezzi, T.J.R. Hughes, L.D. Marini and A. Masud and obtain the computational scheme that is stable at the contrast value of the tensor permeability coefficient. Two hierarchical basis systems for the velocity from Hdiv space and the basis system for pressure from L 2 space are constructed. The verification of the developed computational scheme is conducted on the class of problems which have an analytical solution.

The analysis of the obtained numerical results shows that the inclusion of bubblefunctions in either hierarchical basis systems for velocity allows reducing errors and increasing the smoothness of the solution.

[full text]
Keywords: filtration theory, Darcy model, mixed finite element method, discontinuous Galerkin formulation

doi: 10.5072/ICT.2018.1.11853

Author(s):
Itkina Natalya Borisovna
PhD. , Associate Professor
Position: Associate Professor
Office: Novosibirsk State Technical University
Address: 630073, Russia, Novosibirsk
Phone Office: (383) 346-27-76
E-mail: itkina.nat@yandex.ru

Trofimova Svetlana Alekseevna
Position: Student
Office: Novosibirsk State Technical University, Trofimuk Institute of Petroleum Geology and Geophysics SB RAS
Address: 630073, Russia, Novosibirsk, 20, Prospekt K. Marksa
E-mail: TrofimovaSA@ipgg.sbras.ru

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Bibliography link:
Itkina N.B., Trofimova S.A. Application of mixed non-conforming variational formulations for solving the Darcy problem // Computational technologies. 2018. V. 23. № 1. P. 46-60