Section: New Results

Structure-preserving numerical methods

The design and the analysis of numerical methods preserving at the discrete level the key features of the continuous models is one of the core tasks of the RAPSODI project-team. C. Cancès was invited to write a review paper [16] on energy stable numerical methods for complex porous media flows. The paper addresses three different approaches: monotonicity-based numerical methods like two-point flux approximation Finite Volumes, as well as two methods based on multi-point flow approximation that are either based on upwinding or on positive local dissipation tensors.

Concerning methods based on upwinding, A. Ait Hammou Oulhaj, C. Cancès, and C. Chainais-Hillairet extend in [12] the nonlinear Control Volume Finite Element scheme of [69] to the discretization of Richards equation modeling unsaturated flows in porous media. This strategy is also applied in [30] by A. Ait Hammou Oulhaj and D. Maltese for the simulation of seawater intrusion in the subsoil nearby coastal regions. The scheme proposed in [30] is still convergent if the porous medium is anisotropic, in opposition to the energy-diminishing scheme analyzed in [11] by A. Ait Hammou Oulhaj, which is designed to be accurate in the long-time regime studied in [29]. Besides, an implicit Euler-Finite Volume scheme for a degenerate cross-diffusion system describing the ion transport through biological membranes is analyzed in [17] by C. Cancès, C. Chainais-Hillairet et al. The strongly coupled equations for the ion concentrations include drift terms involving the electric potential, which is coupled to the concentrations through the Poisson equation. The cross-diffusion system possesses a formal gradient flow structure revealing nonstandard degeneracies, which lead to considerable mathematical difficulties. The Finite Volume scheme is based on two-point flux approximations with “double” upwind mobilities. It preserves the structure of the continuous model like non-negativity, upper bounds, and entropy dissipation.

Concerning methods based on positive local dissipation tensors, C. Cancès, C. Chainais-Hillairet et al. propose in [18] a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy/energy dissipation relation. In [37], C. Cancès and co-workers propose a Finite Element scheme for the numerical approximation of degenerate parabolic problems in the form of a nonlinear anisotropic Fokker–Planck equation. The scheme is energy-stable, only involves physically motivated quantities in its definition, and is able to handle general unstructured grids. Its convergence is rigorously proven thanks to compactness arguments, under very general assumptions. Although the scheme is based on Lagrange Finite Elements of degree 1, it is locally conservative after a local post-processing giving rise to an equilibrated flux. This also allows to derive a guaranteed a posteriori error estimate for the approximate solution. Numerical experiments are presented in order to give evidence of a very good behavior of the proposed scheme in various situations involving strong anisotropy and drift terms.

C. Cancès et al. derive in [36] a model of degenerate Cahn–Hilliard type for the phase segregation in incompressible multiphase flows. The model is obtained as the Wasserstein gradient flow of a Ginzburg–Landau energy with the constraint that the sum of the volume fractions must stay equal to 1. The resulting model differs from the classical degenerate Cahn–Hilliard model (see  [106], [85]) and is closely related to a model proposed by E and collaborators  [84], [100]. Besides the derivation of the model, the convergence of a minimizing movement scheme is proven in [36]. The Wasserstein gradient flow structure of the PDE system governing multiphase flows in porous media has recently been highlighted in [68]. The model can thus be approximated by means of the minimizing movement (or JKO) scheme, that C. Cancès et al. solve in [19] thanks to the ALG2-JKO scheme proposed in [60]. The numerical results are compared to a classical upstream mobility Finite Volume scheme, for which strong stability properties can be established.

In [42], S. Lemaire builds a bridge between the Hybrid High-Order [78] and Virtual Element [59] methods, which are the two main new-generation approaches to the arbitrary-order approximation of PDEs on meshes with general, polytopal cells. The Virtual Element method writes in functional terms and is naturally conforming; at the opposite, the Hybrid High-Order method writes in algebraic terms and is naturally nonconforming. It has been remarked a few years ago that the Hybrid High-Order method can be viewed as a nonconforming version of the Virtual Element method. In [42], S. Lemaire ends up unifying the Hybrid High-Order and Virtual Element approaches by showing that the Virtual Element method can be reformulated as a (newborn) conforming Hybrid High-Order method. This parallel has interesting consequences: it allows important simplifications in the a priori analysis of Virtual Element methods, and sheds new light on the differences between conforming and nonconforming Virtual Element methods, in particular in terms of mesh assumptions.

In [31], I. Lacroix-Violet et al. are interested in the numerical integration in time of nonlinear Schrödinger equations using different methods preserving the energy or a discrete analog of it. In particular, they give a rigorous proof of the order of the relaxation method (presented in [62] for cubic nonlinearities) and they propose a generalized version that allows to deal with general power law nonlinearities. Numerical simulations for different physical models show the efficiency of these methods.