This article is about the development and the analysis of an enhanced positive control volume finite element scheme for degenerate convection-diffusion type problems. The proposed scheme involves only vertex unknowns and features anisotropic fields. The novelty of the approach is to devise a reliable upwind approximation with respect to flux-like functions for the elliptic term. Then, it is shown that the discrete solution remains nonnegative. Under general assumptions on the data and the mesh, the convergence of the numerical scheme is established owing to a recent compactness argument. The efficiency and stability of the methodology are numerically illustrated for different anisotropic ratios and nonlinearities.

In this article, a new nodal discretization is proposed for two-phase Darcy flows in heterogeneous porous media. The scheme combines the Vertex Approximate Gradient (VAG) scheme for the approximation of the gradient fluxes with an Hybrid Upwind (HU) approximation of the mobility terms in the saturation equation. The discretization in space incorporates naturally nodal interface degrees of freedom (d.o.f.) allowing to capture the transmission conditions at the interface between different rock types for general capillary pressure curves. It is shown to guarantee the physical bounds for the saturation unknowns as well as a nonnegative lower bound on the capillary energy flux term. Numerical experiments on several test cases exhibit that the scheme is more robust compared with previous approaches allowing the simulation of 3D large Discrete Fracture Matrix (DFM) models. 1 discretizations like the Mixed Hybrid Finite Element (MHFE) method [34,2] or the Hybrid Finite Volume scheme [1]. The additional asset of nodal discretizations compared with cell centered discretizations is to include d.o.f. at rock type interfaces as long as the mesh is conforming with the heterogeneities of the porous medium. These nodal pressures and saturations unknowns will be used to enforce the Darcy normal flux continuity equations combined with the saturation jump condition [16,11,27]. In this work, we will consider the Vertex Approximate Gradient (VAG) discretization developed in [27,12,13] for two-phase Darcy flows in heterogeneous media. The VAG scheme is based on nodal d.o.f. like Control Volume Finite Element (CVFE) methods but it also includes the cell d.o.f. which are eliminated at the linear algebra level at each Newton iteration without any fill-in. These cell d.o.f. provide an additional flexibility in the choice of the control volumes in order to avoid mixing different rock types at nodal control volumes. It has been shown in [27] to be more accurate than usual CVFE discretizations.The time integration will be chosen implicit to avoid severe time step restrictions in high velocity regions such as fractures. It will be also fully coupled to account for the strong coupling between the pressure and saturation unknowns in the transmission conditions at different rock type interfaces. As noticed in [5], the pressure and saturation unknowns cannot be decoupled at such different rock type interfaces to preserve the stability of the discretization.The selected numerical method should also provide a robust nonlinear convergence of the Newton type solver to allow for large time steps, typically at the time scale of the matrix in DFM simulations. Let us refer to [41,37] for alternative linearization schemes to the Newton method which aim to the improvement of the nonlinear solver. In this work, the additional nonlinear solver robustness will be first achieved by extension of the Hybrid Upwind (HU) transport scheme to the VAG discretization framework. The HU transport scheme has been introduced in [24,28] as an alternative to the Phase Pot...

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This work proposes a finite volume scheme for two-phase Darcy flow in heterogeneous porous media with different rock types. The fully implicit discretization is based on cell-centered, as well as face-centered degrees of freedom in order to capture accurately the nonlinear transmission conditions at different rock type interfaces. These conditions play a major role in the flow dynamics. The scheme is formulated with natural physical unknowns, and the notion of global pressure is only introduced to analyze its stability and convergence. It combines a two-point flux approximation of the gradient normal fluxes with a Hybrid Upwinding approximation of the transport terms. The convergence of the scheme to a weak solution is established taking into account the discontinuous capillary pressure at different rock type interfaces and the degeneracy of the phase mobilities. Numerical experiments show the additional robustness of the proposed discretization compared with the classical Phase Potential Upwinding approach.

We are concerned with the approximation of solutions to a compressible two-phase flow model in porous media thanks to an enhanced control volume finite element discretization. The originality of the methodology consists in treating the case where the densities are depending on their own pressures without any major restriction neither on the permeability tensor nor on the mesh. Contrary to the ideas of [23], the point of the current scheme relies on a phaseby-phase "sub"-unpwinding approach so that we can recover the coercivity-like property. It allows on a second place for the preservation of the physical bounds on the discrete saturation. The convergence of the numerical scheme is therefore performed using classical compactness arguments. Numerical experiments are presented to exhibit the efficiency and illustrate the qualitative behavior of the implemented method.

This work addresses the development and analysis of a second-order accurate finite volume scheme for parabolic equations with anisotropy on general simplicial meshes. The discretization involves only vertex unknowns without processing additional ones. The scheme construction makes use of a nonlinear transformation of the linear elliptic term. Two propositions are mainly presented for the approximation of the mobility function at the interfaces. The existence of positive solutions for the discrete system is guaranteed thanks to the proved a priori estimates. The energy dissipation of the scheme is moreover ensured. The convergence of the approach is established. Numerical tests are given to show the efficiency, accuracy and robustness of the proposed approach, with respect to the anisotropy, while a particular emphasis is set on the effects of the approximate mobility. They also confirm the obtained theoretical results, especially the decay of the free energy when time grows.

In this work, we carry out the convergence analysis of a positive DDFV method for approximating solutions of degenerate parabolic equations. The basic idea rests upon different approximations of the fluxes on the same interface of the control volume. Precisely, the approximated flux is split into two terms corresponding to the primal and dual normal components. Then the first term is discretized using a centered scheme whereas the second one is approximated in a non evident way by an upstream scheme. The novelty of our approach is twofold: on the one hand we prove that the resulting scheme preserves the positivity and on the other hand we establish energy estimates. Some numerical tests are presented and they show that the scheme in question turns out to be robust and efficient. The accuracy is almost of second order on general meshes when the solution is smooth.

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