Improving Newton's method performance by parametrization: the case of Richards equation

The nonlinear systems obtained by discretizing degenerate parabolic equations may be hard to solve, especially with Newton's method. In this paper, we apply to Richards equation a strategy that consists in defining a new primary unknown for the continuous equation in order to stabilize Newton's method by parametrizing the graph linking the pressure and the saturation. The resulting form of Richards equation is then discretized thanks to a monotone Finite Volume scheme. We prove the well-posedness of the numerical scheme. Then we show under appropriate non-degeneracy conditions on the parametrization that Newton\^as method converges locally and quadratically. Finally, we provide numerical evidences of the efficiency of our approach

Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure

We consider an immiscible incompressible two-phase flow in a porous medium composed of two different rocks so that the capillary pressure field is discontinuous at the interface between the rocks. This leads us to apply a concept of multi-valued phase pressures and a notion of weak solution for the flow which have been introduced in [Cancés \& Pierre, {\em SIAM J. Math. Anal}, 44(2):966--992, 2012]. We discretize the problem by means of a numerical algorithm which reduces to a standard finite volume scheme in each rock and prove the convergence of the approximate solution to a weak solution of the two-phase flow problem. The numerical experiments show in particular that this scheme permits to reproduce the oil trapping phenomenon

A Trefftz-like coarse space for the two-level Schwarz method on perforated domains

We consider a new coarse space for the ASM and RAS preconditioners to solve elliptic partial differential equations on perforated domains, where the numerous polygonal perforations represent structures such as walls and buildings in urban data. With the eventual goal of modelling urban floods by means of the nonlinear Diffusive Wave equation, this contribution focuses on the solution of linear problems on perforated domains. Our coarse space uses a polygonal subdomain partitioning and is spanned by Trefftz-like basis functions that are piecewise linear on the boundary of a subdomain and harmonic inside it. It is based on nodal degrees of freedom that account for the intersection between the perforations and the subdomain boundaries. As a reference, we compare this coarse space to the well-studied Nicolaides coarse space with the same subdomain partitioning. It is known that the Nicolaides space is unable to prevent stagnation in convergence when the subdomains are not connected; we work around this issue by separating each subdomain by disconnected component. Scalability and robustness are tested for multiple data sets based on realistic urban topography. Numerical results show that the new coarse space is very robust and accelerates the number of Krylov iterations when compared to Nicolaides, independent of the complexity of the data.Comment: 9 pages, 4 figures, submitted to the 27th International Conference on Domain Decomposition Methods proceeding

Robust Methods for Multiscale Coarse Approximations of Diffusion Models in Perforated Domains

For the Poisson equation posed in a domain containing a large number of polygonal perforations, we propose a low-dimensional coarse approximation space based on a coarse polygonal partitioning of the domain. Similarly to other multiscale numerical methods, this coarse space is spanned by locally discrete harmonic basis functions. Along the subdomain boundaries, the basis functions are piecewise polynomial. The main contribution of this article is an error estimate regarding the H1-projection over the coarse space which depends only on the regularity of the solution over the edges of the coarse partitioning. For a specific edge refinement procedure, the error analysis establishes superconvergence of the method even if the true solution has a low general regularity. Combined with domain decomposition (DD) methods, the coarse space leads to an efficient two-level iterative linear solver which reaches the fine-scale finite element error in few iterations. It also bodes well as a preconditioner for Krylov methods and provides scalability with respect to the number of subdomains. Numerical experiments showcase the increased precision of the coarse approximation as well as the efficiency and scalability of the coarse space as a component of a DD algorithm.Comment: 32 pages, 14 figures, submitted to Journal of Computational Physic

Coupling of a two phase gas liquid compositional 3D Darcy flow with a 1D compositional free gas flow

International audienceA model coupling a three dimensional gas liquid compositional Darcy flow and a one dimensional compositional free gas flow is presented. The coupling conditions at the interface between the gallery and the porous media account for the molar normal fluxes continuity for each component, the gas liquid thermodynami-cal equilibrium, the gas pressure continuity and the gas and liquid molar fractions continuity. This model is applied to the simulation of the mass exchanges at the interface between the repository and the ventilation excavated gallery in a nuclear waste geological repository. The convergence of the Vertex Approximate Gradient discretization is analysed for a simplified model coupling the Richards approximation in the porous media and the gas pressure equation in the gallery

Convergence of a Vertex centred Discretization of Two-Phase Darcy flows on General Meshes

International audienceThis article analyses the convergence of the Vertex Approximate Gradient (VAG) scheme recently introduced for the discretization of multiphase Darcy flows on general polyhedral meshes. The convergence of the scheme to a weak solution is shown in the particular case of an incompressible immiscible two phase Darcy flow model with capillary diffusion using a global pressure formulation. A remarkable property in practice is that the convergence is proven whatever the distribution of the volumes at the cell centres and at the vertices used in the control volume discretization of the saturation equation. The numerical experiments carried out for various families of 2D and 3D meshes confirm this result on a one dimensional Buckley Leverett solution

Convergence of a Vertex centred Discretization of Two-Phase Darcy flows on General Meshes

International audienceThis article analyses the convergence of the Vertex Approximate Gradient (VAG) scheme recently introduced for the discretization of multiphase Darcy flows on general polyhedral meshes. The convergence of the scheme to a weak solution is shown in the particular case of an incompressible immiscible two phase Darcy flow model with capillary diffusion using a global pressure formulation. A remarkable property in practice is that the convergence is proven whatever the distribution of the volumes at the cell centres and at the vertices used in the control volume discretization of the saturation equation. The numerical experiments carried out for various families of 2D and 3D meshes confirm this result on a one dimensional Buckley Leverett solution

Don't Trust Your Eyes: Image Manipulation in the Age of DeepFakes

We review the phenomenon of deepfakes, a novel technology enabling inexpensive manipulation of video material through the use of artificial intelligence, in the context of today’s wider discussion on fake news. We discuss the foundation as well as recent developments of the technology, as well as the differences from earlier manipulation techniques and investigate technical countermeasures. While the threat of deepfake videos with substantial political impact has been widely discussed in recent years, so far, the political impact of the technology has been limited. We investigate reasons for this and extrapolate the types of deepfake videos we are likely to see in the future.publishedVersio

COVID-19 and 5G conspiracy theories: long term observation of a digital wildfire

The COVID-19 pandemic has severely affected the lives of people worldwide, and consequently, it has dominated world news since March 2020. Thus, it is no surprise that it has also been the topic of a massive amount of misinformation, which was most likely amplified by the fact that many details about the virus were not known at the start of the pandemic. While a large amount of this misinformation was harmless, some narratives spread quickly and had a dramatic real-world effect. Such events are called digital wildfires. In this paper we study a specific digital wildfire: the idea that the COVID-19 outbreak is somehow connected to the introduction of 5G wireless technology, which caused real-world harm in April 2020 and beyond. By analyzing early social media contents we investigate the origin of this digital wildfire and the developments that lead to its wide spread. We show how the initial idea was derived from existing opposition to wireless networks, how videos rather than tweets played a crucial role in its propagation, and how commercial interests can partially explain the wide distribution of this particular piece of misinformation. We then illustrate how the initial events in the UK were echoed several months later in different countries around the world.publishedVersio