Iuliu Sorin Pop

Florin Adrian Radu, Kundan Kumar, Jan Martin Nordbotten et al.

In this work we present a mass conservative numerical scheme for two-phase flow in porous media. The model for flow consists on two fully coupled, non-linear equations: a degenerate parabolic equation and an elliptic equation. The proposed numerical scheme is based on backward Euler for the temporal discretization and mixed finite element method (MFEM) for the discretization in space. Continuous, semi-discrete (continuous in space) and fully discrete variational formulations are set up and the existence and uniqueness of solutions is discussed. Error estimates are presented to prove the convergence of the scheme. The non-linear systems within each time step are solved by a robust linearization method. This iterative method does not involve any regularization step. The convergence of the linearization scheme is rigorously proved under the assumption of a Lipschitz continuous saturation. The case of a Hölder continuous saturation is also discussed, a rigorous convergence proof being given for Richards' equation. Numerical results are presented to sustain the theoretical findings.

Jakub W. Both, Kundan Kumar, Jan M. Nordbotten et al.

Mathematical models for flow and reactive transport in porous media often involve non-linear, degenerate parabolic equations. Their solutions have low regularity, and therefore lower order schemes are used for the numerical approximation. Here the backward Euler method is combined with a mixed finite element method scheme, which results in a stable and locally mass-conservative scheme. At the same time, at each time step one has to solve a non-linear algebraic system, for which linear iterations are needed. Finding robust and convergent ones is particularly challenging here, since both slow and fast diffusion cases are allowed. Commonly used schemes, like Newton and Picard iterations, are defined either for non-degenerate problems, or after regularising the problem in the case of degenerate ones. Convergence is guaranteed only if the initial guess is sufficiently close to the solution, which translates into severe restrictions on the time step. Here we discuss a linear iterative scheme which builds on the $L$-scheme, and does not employ any regularisation. We prove its rigourous convergence, which is obtained for mild restrictions on the time step. Finally, we give numerical results confirming the theoretical ones, and compare the behaviour of the scheme with other schemes.

David Seus, Koondanibha Mitra, Iuliu Sorin Pop et al.

The Richards equation is a nonlinear parabolic equation that is commonly used for modelling saturated/unsaturated flow in porous media. We assume that the medium occupies a bounded Lipschitz domain partitioned into two disjoint subdomains separated by a fixed interface $Γ$. This leads to two problems defined on the subdomains which are coupled through conditions expressing flux and pressure continuity at $Γ$. After an Euler implicit discretisation of the resulting nonlinear subproblems a linear iterative ($L$-type) domain decomposition scheme is proposed. The convergence of the scheme is proved rigorously. In the last part we present numerical results that are in line with the theoretical finding, in particular the unconditional convergence of the scheme. We further compare the scheme to other approaches not making use of a domain decomposition. Namely, we compare to a Newton and a Picard scheme. We show that the proposed scheme is more stable than the Newton scheme while remaining comparable in computational time, even if no parallelisation is being adopted. Finally we present a parametric study that can be used to optimize the proposed scheme.