Abstract
We discuss two linearization and domain decomposition methods for mathematical models for two-phase flow in a porous medium. The medium consists of two adjacent regions with possibly different parameterizations. The model accounts for non-equilibrium effects like dynamic capillarity and hysteresis. The θ-scheme is adopted for the temporal discretization of the equations yielding nonlinear time-discrete equations. For these, we propose and analyze two iterative schemes, which combine a stabilized linearization iteration of fixed-point type, the L-scheme, and a non-overlapping domain decomposition method into one iteration. First, we prove the existence of unique solutions to the problems defining the linear iterations. Then, we give the rigorous convergence proof for both iterative schemes towards the solution of the time-discrete equations. The developed schemes are independent of the spatial discretization or the mesh and avoid the use of derivatives as in Newton based iterations. Their convergence holds independently of the initial guess, and under mild constraints on the time step. The numerical examples confirm the theoretical results and demonstrate the robustness of the schemes. In particular, the second scheme is well suited for models incorporating hysteresis. The schemes can be easily implemented for realistic applications.
Original language | English |
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Article number | 113364 |
Number of pages | 27 |
Journal | Computer Methods in Applied Mechanics and Engineering |
Volume | 372 |
DOIs | |
Publication status | Published - 1 Dec 2020 |
Funding
This work was supported by the Hasselt University (Project BOF17NI01) and the Research Foundation Flanders (FWO, Project G051418N). We also would like to thank the anonymous referees for their careful reading and constructive suggestions to improve the paper.
Keywords
- Domain decomposition
- Dynamic capillarity
- Hysteresis
- Linearization
- Two-phase flow in porous media