## June 6 – Quanling Deng: High-order generalized-alpha methods and splitting schemes

Quanling Deng: Thursday 6 June at 11:00, A415 Inria Paris. The well-known generalized-alpha method is an unconditionally stable and second-order accurate time-integrator which has a feature of user-control on numerical dissipation. The method encompasses a wide range of time-integrators, such as the Newmark method, the HHT-alpha method by Hilber, Hughes, and Taylor, and the WBZ-alpha method by Wood, Bossak, and Zienkiewicz. The talk starts with the simplest time-integrator, forward/backward Euler schemes, then introduces Newmark’s idea followed by the ideas of Chung and Hulbert on the generalized-alpha method. For parabolic equations, we show that the generalized-alpha method also includes the BDF-2 and the second-order dG time-integration scheme. The focus of the talk is to introduce two ideas to generalize the method further to higher orders while maintaining the features of unconditional stability and dissipation control. We will show third-order (for parabolic equations) and 2n-order (for hyperbolic equations) accurate schemes with numerical validations. The talk closes with the introduction of a variational-splitting framework for these time-integrators. As a consequence, the splitting schemes reduce the computational costs significantly (to linear cost) for multi-dimensional problems.

## April 12 – Menel Rahrah: Mathematical modelling of fast, high volume infiltration in poroelastic media using finite elements

Menel Rahrah: Friday 12 April at 14:30, A315 Inria Paris. Fast, High Volume Infiltration (FHVI) is a new method to quickly infiltrate large amounts of fresh water into the shallow subsurface. This infiltration method would have a great value for the effective storage of rainwater in the underground, during periods of extreme precipitation. To describe FHVI, a model for aquifers is considered in which water is injected. Water injection induces changes in the pore pressure and deformations in the soil. Furthermore, the interaction between the mechanical deformations and the flow of water gives rise to a change in porosity and permeability, which results in nonlinearity of the mathematical problem. Assuming that the deformations are very small, Biot’s theory of linear poroelasticity is used to determine the local displacement of the skeleton of a porous medium, as well as the fluid flow through the pores. The resulting problem needs a considerate numerical methodology in terms of possible nonphysical oscillations. Therefore, a stabilised Galerkin finite element method based on Taylor-Hood elements is developed. Subsequently, the impact of mechanical oscillations and pressure pulses on the amount of water that can be injected into the aquifer is investigated. In addition, a parameter uncertainty quantification is applied using Monte Carlo techniques and statistical analysis, to quantify the impact of variation in the parameters (such as the unknown oscillatory modes and the soil characteristics) on the model output. Since the assumption that the deformations are very small can be violated by imposing large mechanical oscillations, the difference between the linear and the nonlinear poroelasticity equations is analysed in a moving finite element framework using Picard iterations.

## March 18 – Patrik Daniel: Adaptive hp-finite elements with guaranteed error contraction and inexact multilevel solvers

Patrik Daniel: Monday 18 March at 14:00, A315 Inria Paris. We propose new practical adaptive refinement algorithms for conforming hp-finite element approximations of elliptic problems. We consider the use of both exact and inexact solvers within the established framework of adaptive methods consisting of four concatenated modules: SOLVE, ESTIMATE, MARK, REFINE. The strategies are driven by guaranteed equilibrated flux a posteriori error estimators. Namely, for an inexact approximation obtained by an (arbitrary) iterative algebraic solver, the bounds for the total, the algebraic, and the discretization errors are provided. The nested hierarchy of hp-finite element spaces is crucially exploited for the algebraic error upper bound which in turn allows us to formulate sharp stopping criteria for the algebraic solver. Our hp-refinement criterion hinges on from solving two local residual problems posed on patches of elements around marked vertices selected by a bulk-chasing criterion. They respectively emulate h-refinement and p-refinement. One particular feature of our approach is that we derive a computable real number which gives a guaranteed bound on the ratio of the (unknown) energy error in the next adaptive loop step with respect to the present one (i.e. on the error reduction factor). Numerical experiments are presented to validate the proposed adaptive strategies. We investigate the accuracy of our bound on the error reduction factor which turns out to be excellent, with effectivity indices close to the optimal value of one. In practice, we observe asymptotic exponential convergence rates, in both the exact and inexact algebraic solver settings. Finally, we also provide a theoretical analysis of the proposed strategies. We prove that under some additional assumptions on the h- and p-refinements, including the interior node property and sufficient p-refinements, the computable reduction factors are indeed bounded by a generic constant strictly smaller than one. This implies the convergence of the…

## February 14 – Thibault Faney, Soleiman Yousef: Accélération d’un simulateur d’équilibres thermodynamiques par apprentissage automatique

Thibault Faney, Soleiman Yousef: Thursday 14 February at 15:00, A415 Inria Paris. Les simulations numériques sont un outil important pour mieux comprendre et prédire le comportement des systèmes physiques complexes. Dans le cas des équilibres thermodynamiques, ces systèmes sont modélisés principalement par des systèmes d’équations non-linéaires. La résolution de ces systèmes après discrétisation nécessite l’emploi de méthodes itératives se basant sur une solution initiale approximative. L’objectif des travaux présentés est d’établir un modèle statistique par apprentissage permettant de remplacer cette solution initiale heuristique par une approximation de la solution fondée sur une base de données de résultats de calculs. Le simulateur ciblé est un simulateur de flash triphasique (aqueux, liquide et gaz) à pression et température constante. A partir de fractions molaires initiales des éléments chimiques, on calcule les phases présentes à l’état d’équilibre, ainsi que leur fraction volumique respective et les fractions molaires de chaque espèce chimique dans chacune des phases présentes. Nous formulons le problème sous la forme d’un problème d’apprentissage supervisé en série : d’abord une classification des phases présentes à l’équilibre puis une régression sur les fractions volumiques et molaires. Les résultats sur différents tests de complexité croissante en nombre de composés chimiques montrent un très bon accord du modèle statistique avec la solution fournie par le simulateur. Les éléments clés sont le choix de la base de données pour l’apprentissage (plan d’expérience) et l’emploi de méthodes d’apprentissage non-linéaires.

## January 31 – Camilla Fiorini: Sensitivity analysis for hyperbolic PDEs systems with discontinuous solution: the case of the Euler Equations.

Camilla Fiorini: Thursday 31 January at 14:30, A415 Inria Paris. Sensitivity analysis (SA) is the study of how changes in the input of a model affect the output. Standard SA techniques for PDEs, such as the continuous sensitivity equation method, call for the differentiation of the state variable. However, if the governing equations are hyperbolic PDEs, the state can be discontinuous and this generates Dirac delta functions in the sensitivity. We aim at defining and approximating numerically a system of sensitivity equations which is valid also when the state is discontinuous: to do that, one can define a correction term to be added to the sensitivity equations starting from the Rankine-Hugoniot conditions, which govern the state across a shock. We detail this procedure in the case of the Euler equations. Numerical results show that the standard Godunov and Roe schemes fail in producing good numerical results because of the underlying numerical diffusion. An anti-diffusive numerical method is then successfully proposed.