# Internal Seminar

## 2020 – 2021

• 17 June 2021 – 11h00 to 12h00: Elyes Ahmed: Adaptive fully-implicit solvers and a posteriori error control for multiphase flow with wells. (abstract)
• 3 June 2021 – 11h00 to 12h00: Oliver Sutton: High order, mesh-based multigroup discrete ordinates schemes for the linear Boltzmann transport problem. (abstract)
• 29 April 2021 – 11h00 to 12h00: Lorenzo Mascotto: Enriched nonconforming virtual element methods (abstract)
• 1 April 2021 – 11h00 to 12h00: André Harnist : Improved error estimates for Hybrid High-Order discretizations of Leray–Lions problems (abstract)
• 11 March 2021 – 15h00 to 16h00: Omar Duran : Explicit and implicit hybrid high-order methods for the wave equation in time regime (abstract)
• 25 February 2021 – 14h00 to 15h00: Buyang Li : A bounded numerical solution with a small mesh size implies existence of a smooth solution to the time-dependent Navier–Stokes equations (abstract)
• 18 February 2021 – 11h00 to 12h00: Roland Maier :  Multiscale scattering in nonlinear Kerr-type media (abstract)
• 10 December 2020 – 16h00 to 17h00: Ani Miraçi : A-posteriori-steered and adaptive p-robust multigrid solvers (abstract)
• 9 December 2020 – 16h00 to 17h00: Riccardo Milani : Compatible Discrete Operator schemes for the unsteady incompressible Navier–Stokes equations (abstract)
• 26 November 2020 – 16h00 to 17h00: Koondanibha Mitra : A posteriori error bounds for the Richards equation (abstract)
• 19 November 2020 – 11h00 to 12h00: Joëlle Ferzly : Semismooth and smoothing Newton methods for nonlinear systems with complementarity constraints: adaptivity and inexact resolution (abstract)
• 5 November 2020 – 11h00 to 12h00: Zhaonan Dong : On a posteriori error estimates for non-conforming Galerkin methods (abstract)
• 22 October 2020 – 11h00 to 12h00: Théophile Chaumont-Frelet : A posteriori error estimates for Maxwell’s equations based on flux quasi-equilibration (abstract)
• 15 October 2020 – 11h00 to 12h00: Florent Hédin : A hybrid high-order (HHO) method with non-matching meshes in discrete fracture networks (abstract)

## 2019 – 2020

• 16 March 2020 – 15h00 to 16h00: Bochra Mejri : Topological sensitivity analysis for identification of voids under Navier’s boundary conditions in linear elasticity (abstract)
• 25 February 2020 – 15h00 to 16h00: Jakub Both : Robust iterative solvers for thermo-poro-visco-elasticity via gradient flows (abstract)
• 16 October 2019 – 14h00 to 15h00: Nicolas Pignet : Hybrid High-Order method for nonlinear solid mechanics (abstract)
• 27 September 2019 – 15h00 to 16h00: Ivan Yotov : A nonlinear Stokes-Biot model for the interaction of a non-Newtonian fluid with poroelastic media (abstract)
• 5 September 2019 – 15h00 to 16h00: Koondi Mitra : A fast and stable linear iterative scheme for nonlinear parabolic problems (abstract)

## 2018 – 2019

• 11 July 2019 – 11h00 to 12h00: Jose Fonseca : Towards scalable parallel adaptive simulations with ParFlow (abstract)
• 6 June 2019 – 11h00 to 12h00: Quanling Deng : High-order generalized-alpha methods and splitting schemes (abstract)
• 12 April 2019 – 14h30 to 15h30: Menel Rahrah : Mathematical modelling of fast, high volume infiltration in poroelastic media using finite elements (abstract)
• 18 March 2019 – 14h to 15h: Patrik Daniel : Adaptive hp-finite elements with guaranteed error contraction and inexact multilevel solvers (abstract)
• 14 February 2019 – 15h to 16h: Thibault Faney, Soleiman Yousef : Accélération d’un simulateur d’équilibres thermodynamiques par apprentissage automatique (abstract)
• 7 February 2019 – 11h to 12h: Gregor Gantner : Optimal adaptivity for isogeometric finite and boundary element methods (abstract)
• 31 January 2019 – 14h30 to 15h30: Camilla Fiorini : Sensitivity analysis for hyperbolic PDEs systems with discontinuous solution: the case of the Euler Equations. (abstract)
• 9 January 2019 – 11h to 12h: Zhaonan Dong : hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes (abstract)
• 13 December 2018 – 11h to 12h: Maxime Breden : An introduction to a posteriori validation techniques, illustrated on the Navier-Stokes equations (abstract)
• 5 December 2018 – 11h00 to 12h00: Amina Benaceur : Model reduction for nonlinear thermics and mechanics (abstract)

## 2017 – 2018

• 16 April 2018 – 15h to 16h: Simon Lemaire : An optimization-based method for the numerical approximation of sign-changing PDEs (abstract)
• 20 Febraury 2018 – 15h to 16h: Thirupathi Gudi : An energy space based approach for the finite element approximation of the Dirichlet boundary control problem (abstract)
• 15 Febraury 2018 – 14h to 15h: Franz Chouly : About some a posteriori error estimates for small strain elasticity (abstract)
• 30 November 2017 – 14h to 15h: Sébastien Furic : Construction & Simulation of System-Level Physical Models (abstract)
• 2 November 2017 – 11h to 12h: Hend Benameur: Identification of parameters, fractures ans wells in porous media (abstract)
• 10 October 2017 – 11h to 12h: Peter Minev: Recent splitting schemes for the incompressible Navier-Stokes equations (abstract)
• 18 September 2017 – 13h to 14h: Théophile Chaumont: High order finite element methods for the Helmholtz equation in highly heterogeneous media (abstract)

## 2016 – 2017

• 29 June 2017 – 15h to 16h: Gouranga Mallik: A priori and a posteriori error control for the von Karman equations (abstract)
• 22 June 2017 – 15h to 16h: Valentine Rey: Goal-oriented error control within non-overlapping domain decomposition methods to solve elliptic problems (abstract)
• 15 June 2017 – 15h to 16h:
• 6 June 2017 – 11h to 12h: Ivan Yotov: Coupled multipoint flux and multipoint stress mixed finite element methods for poroelasticity (abstract)
• 1 June 2017 – 10h to 12h:
• Joscha GedickeAn adaptive finite element method for two-dimensional Maxwell’s equations (abstract)
• Martin EigelAdaptive stochastic FE for explicit Bayesian inversion with hierarchical tensor representations (abstract)
• Quang Duc Bui: Coupled Parareal-Schwarz Waveform relaxation method for advection reaction diffusion equation in one dimension (abstract)
• 16 May 2017 – 15h to 16h: Quanling Deng: Dispersion Optimized Quadratures for Isogeometric Analysis (abstract)
• 11 May 2017 – 15h to 16h: Sarah Ali Hassan: A posteriori error estimates and stopping criteria for solvers using domain decomposition methods and with local time stepping (abstract)
• 13 Apr. 2017 – 15h to 16h: Janelle Hammond: A non intrusive reduced basis data assimilation method and its application to outdoor air quality models (abstract)
• 30 Mar. 2017 – 10h to 11h: Mohammad Zakerzadeh: Analysis of space-time discontinuous Galerkin scheme for hyperbolic and viscous conservation laws (abstract)
• 23 Mar. 2017 – 15h to 16h: Karol Cascavita: Discontinuous Skeletal methods for yield fluids (abstract)
• 16 Mar. 2017 – 15h to 16h: Thomas Boiveau: Approximation of parabolic equations by space-time tensor methods (abstract)
• 9 Mar. 2017 – 15h to 16h: Ludovic Chamoin: Multiscale computations with MsFEM: a posteriori error estimation, adaptive strategy, and coupling with model reduction (abstract)
• 2 Mar. 2017 – 15h to 16h: Matteo Cicuttin: Implementation of Discontinuous Skeletal methods on arbitrary-dimensional, polytopal meshes using generic programming. (abstract)
• 23 Feb. 2017
10h to 10h45 : Lars Diening: Linearization of the p-Poisson equation (abstract)
10h45 to 11h30 : Christian Kreuzer: Quasi-optimality of discontinuous Galerkin methods for parabolic problems (abstract)
• 26 Jan. 2017 – 15h to 16h: Amina BenaceurAn improved reduced basis method for non-linear heat transfer (abstract)
• 19 Jan. 2017 – 15h to 16h: Laurent Monasse: A 3D conservative coupling between a compressible flow and a fragmenting structure (abstract)
• 5 Jan. 2017 – 15h to 16h: Agnieszka Miedlar: Moving eigenvalues and eigenvectors by simple perturbations (abstract)
• 8 Dec. 2016 – 15h to 16h: Luca Formaggia: Hybrid dimensional Darcy flow in fractured porous media, some recent results on mimetic discretization (abstract)
• 22 Sept. 2016 – 15h to 16h: Paola AntoniettiFast solution techniques for high order Discontinuous Galerkin methods (abstract)

## 2015 – 2016

• 29 Oct. 2015 – 15h to 16h: Sarah Ali HassanA posteriori error estimates for domain decomposition methods (abstract)
• 05 Nov. 2015 – 16h to 17h: Iain SmearsRobust and efficient preconditioners for the discontinuous Galerkin time-stepping method (abstract)
• 12 Nov. 2015 -16h to 17h: Elyes Ahmed: Space-time domain decomposition method for two-phase flow equations (abstract)
• 19 Nov. 2015 – 16h to 17h: Géraldine PichotGeneration algorithms of stationary Gaussian random fields (abstract)
• 26 Nov. 2015-16h to 17h: Jérôme JaffréDiscrete reduced models for flow in porous media with fractures and barriers (abstract)
• 03 Dec. 2015 – 16h to 17h: François Clément: Safe and Correct Programming for Scientific Computing (abstract)
• 10 Dec. 2015 – 16h to 17h: Nabil Birgle: Composite Method on Polygonal Meshes (abstract)11 Feb. 2016: Michel
• Kern: Reactive transport in porous media: Formulations and numerical methods
• 25 Feb. 2016: Martin Vohralík
• 3 March 2016: François Clément: Safe and Correct Programming for Scientific Computing pt II

## 17 June – Elyes Ahmed: Adaptive fully-implicit solvers and a posteriori error control for multiphase flow with wells

Elyes Ahmed: Thursday 17 June at 11:00 am   ABSTRACT: Flow is driven by the wells in most reservoir simulation workflows. From a numerical point of view,  wells can be seen as singular source-terms due to their small-scale relative to grid blocks used in field-scale simulation. Near-well models, such as Peaceman model, are used to account for the highly non-linear flow field in the vicinity of the wellbore. The singularities that wells introduce in the solution create difficulties for the gridding strategy and usually result in a less flexible time-stepping strategy to ensure convergence of the nonlinear solver. We present in this work a-posteriori error estimators for multiphase flow with singular well sources. The estimators are fully and locally computable and target the singular effects of wells.  The error estimate uses the appropriate weighted norms, where the weight weakens the norm only around the wells, letting it behave like the usual H^{1} -norm far from the near-well region. The error estimators are used to modify a fully implicit solver in the MATLAB Reservoir Simulation Toolbox (MRST). We demonstrate the benefits of the adaptive implicit solver through a range of test cases.

## 3 June – Oliver Sutton: High order, mesh-based multigroup discrete ordinates schemes for the linear Boltzmann transport problem.

Oliver Sutton: Thursday 3 June at 11:00 am   ABSTRACT: The linear Boltzmann problem is a widely used model for the transport of particles through a scattering medium, such as neutrons in a nuclear reactor, or photons during radiotherapy or in the atmosphere of a star. The key challenge in simulating such phenomena using this model lies in the fact that the problem is an integro-partial-differential equation in 6 independent variables: three position variables and three momentum variables (7 if time is also included). Despite this, there is a long history of this model being successfully applied in practice. A well-studied class of numerical schemes for simulating phenomena governed by this model couples a discontinuous Galerkin spatial discretisation with a multigroup discrete-ordinates discretisation in the momentum variables. Standard multigroup discrete ordinates discretisations may be viewed as employing a piecewise constant approximation space, and possess the particularly attractive characteristic of decoupling the fully-coupled problem into a sequence of three-dimensional linear transport problems which may be solved independently and in parallel. In this talk, we will discuss a new generalisation of these multigroup discrete ordinates schemes. These new schemes employ arbitrarily high order polynomials in the discretisation of the momentum variables, providing high order convergence properties, and offer a familiar Galerkin framework for their analysis. Crucially, moreover, they retain the simple algorithmic structure of their classical counterparts.

## 29 April – Lorenzo Mascotto: Enriched nonconforming virtual element methods

Lorenzo Mascotto: Thursday 29 April at 11:00 am ABSTRACT: Solutions to elliptic partial differential equations (PDEs) on smooth domains with smooth data are smooth. However, when solving a PDE on a Lipschitz polygon, its solution is singular at the vertices of the domain. The singular behaviour is known a priori: the solution can be split as a sum of a smooth term, plus series of singular terms that belong to the kernel of the differential operator appearing in the PDE. The virtual element method (VEM) is a generalization of the finite element method (FEM) to polygonal/polyhedral meshes and is based on approximation spaces consisting of solutions to local problems mimicking the target PDE and has been recently generalized to the extended VEM (XVEM). Here, the approximation spaces are enriched with suitable singular functions. A partition of unity (PU) is used to patch local spaces. The approach of the XVEM is close to that of the extended FEM. In this talk, we present a new paradigm for enriching virtual element (VE) spaces. Instead of adding special functions to the global space and eventually patch local spaces with a PU, we modify the definition of the local spaces by tuning the boundary conditions of local problems. By doing this, local VE spaces contain the desired singular functions, but there is no need to patch them with a PU. This results in an effective and slight modification of the already existing implementation of VE codes, as well as in a natural extension for existing theoretical results.

## 1 April – André Harnist: Improved error estimates for Hybrid High-Order discretizations of Leray–Lions problems

André Harnist: Thursday 1 April at 11:00 am Abstract: We consider Hybrid High-Order (HHO) approximations of Leray-Lions problems set in W^(1,p) with p in (1,2]. For this class of problems, negative powers of the gradient of the solution can appear in the flux. Depending on the expression of the latter, this can lead to a degeneracy of the problem when the gradient of the solution vanishes or becomes large. The goal of this presentation is to derive novel error estimates depending on the degeneracy of the problem inspired by [1,2,3]. Specifically, we show that, for the globally non-degenerate case, the energy-norm of the error has a convergence rate of (k+1), with k denoting the degree of the HHO approximation. In the globally degenerate case, on the other hand, the energy-norm of the error converges with a rate of (k+1)(p-1), coherently with the estimate originally proved in [4]. We additionally introduce, for each mesh element, a dimensionless number that captures the local degeneracy of the model and identifies the contribution of the element to the global error: from the fully degenerate regime, corresponding to a contribution in (k+1)(p-1), to the non-degenerate regime, corresponding to a contribution in (k+1), through all intermediate regimes. These regime-dependent error estimates are illustrated by a complete panel of numerical experiments. [1] M. Botti, D. Castanon Quiroz, D. A. Di Pietro and A. Harnist, A Hybrid High-Order method for creeping flows of non-Newtonian fluids, Submitted. 2020. URL: https://hal.archives-ouvertes.fr/hal-02519233. [2] D. A. Di Pietro and J. Droniou, The Hybrid High-Order Method for Polytopal Meshes. Modelling, Simulation and Application 19. Springer International Publishing, 2020. ISBN: 978-3-030-37202-6 (Hardcover) 978-3-030-37203-3 (eBook). DOI: 10.1007/978-3-030-37203-3. [3] D. A. Di Pietro and J. Droniou, A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes, Math. Comp., volume 86, 2017, number 307, pages 2159–2191,…

## 11 March – Omar Duran: Explicit and implicit hybrid high-order methods for the wave equation in time regime

Omar Duran: Thursday 11 March at 15:00 There are two main approaches to derive a fully discrete method for solving the second-order acoustic wave equation in the time regime. One is to discretize directly the second-order time derivative and the Laplacian operator in space. The other approach is to transform the second-order equation into a first-order hyperbolic system. Firstly, we consider the time second-order form. We devise, analyze the energy-conservation properties, and evaluate numerically a hybrid high-order (HHO) scheme for the space discretization combined with a Newmark-like time-marching scheme. The HHO method uses as discrete unknowns cell- and face-based polynomials of some order 0 ≤ k, yielding for steady problems optimal convergence of order (k + 1) in the energy norm [1]. Secondly, inspired by ideas presented in [2] for hybridizable discontinuous Galerkin (HDG) method and the link between HDG and HHO methods in the steady case [3], first-order explicit or implicit time-marching schemes combined with the HHO method for space discretization are considered. We discuss the selection of the stabilization term and energy conservation and present numerical examples. Extension to the unfitted meshes is contemplated for the acoustic wave equation. We observe that the unfitted approach combined with local cell agglomeration leads to a comparable CFL condition as when using fitted meshes [4]. [1] D.A. Di Pietro, A. Ern, and S. Lemaire. An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Computational Methods in Applied Mathematics. 14 (2014) 461-472. [2] M. Stanglmeier, N.C. Nguyen, J. Peraire, and B. Cockburn. An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation. Computer Methods in Applied Mechanics and Engineering, 300:748–769, March 2016. [3] B. Cockburn, D. A. Di Pietro, and A. Ern. Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM: Mathematical Modelling and…

## 25 February – Buyang Li : A bounded numerical solution with a small mesh size implies existence of a smooth solution to the time-dependent Navier–Stokes equations

Buyang Li: Thursday 25 February at 14:00 We prove that for a given smooth initial value, if one finite element solution of the three-dimensional time-dependent Navier–Stokes equations is bounded by $M$ when some sufficiently small step size $\tau < \tau _M$ and mesh size $h < h_M$ are used, then the true solution of the Navier–Stokes equations with this given initial value must be smooth and unique, and is successfully approximated by the numerical solution.

## 18 February – Roland Maier : Multiscale scattering in nonlinear Kerr-type media

18 February – Roland Maier Wave propagation in heterogeneous and nonlinear media has arisen growing interest in the last years since corresponding materials can lead to unusual and interesting effects and therefore come with a wide range of applications. An important example of such materials is Kerr-type media, where the intensity of a wave directly influences the refractive index. In the time-harmonic regime, this effect can be modeled with a nonlinear Helmholtz equation. If underlying material coefficients are highly oscillatory on a microscopic scale, the numerical approximation of corresponding solutions can be a delicate task. In this talk, a multiscale technique is presented that allows one to deal with microscopic coefficients in a nonlinear Helmholtz equation without the need for global fine-scale computations. The method is based on an iterative and adaptive construction of appropriate multiscale spaces based on the multiscale method known as Localized Orthogonal Decomposition, which works under minimal structural assumptions. This talk is based on joint work with Barbara Verfürth (KIT, Karlsruhe)

## December 10 – Ani Miraçi: A-posteriori-steered and adaptive p-robust multigrid solvers

Ani Miraçi: Thursday 10 December at 16:00 via this link with meeting ID: 950 9381 2553 and code: 077683 We consider systems of linear algebraic equations arising from discretizations of second-order elliptic partial differential equations using finite elements of arbitrary polynomial degree. First, we propose an a posteriori estimator of the algebraic error, the construction of which is linked to that of a multigrid solver without pre-smoothing and a single post-smoothing step by overlapping Schwarz methods (block-Jacobi). We prove the following results and their equivalence: the solver contracts the algebraic error independently of the polynomial degree (p-robustness); the estimator represents a two-sided p-robust bound of the algebraic error. Next, we introduce optimal step-sizes which minimize the error at each level. This makes it possible to have an explicit Pythagorean formula for the algebraic error; this in turn serves as the foundation for a simple and efficient adaptive strategy allowing to choose the number of post-smoothing steps per level. We also introduce an adaptive local smoothing strategy thanks to our efficient and localized estimator of the algebraic error by levels/patches of elements. Patches contributing more than a user-prescribed percentage to the overall error are marked via a bulk-chasing criterion. Each iteration is composed of a non-adaptive V-cycle and an adaptive V-cycle which uses local smoothing only in the marked patches. These V-cycles contract the algebraic error in a p-robust fashion. Finally, we also extend some the above results to a mixed finite elements setting in two space dimensions. A variety of numerical tests is presented to confirm our theoretical results and to illustrate the advantages of our approaches.

## December 9 – Riccardo Milani: Compatible Discrete Operator schemes for the unsteady incompressible Navier–Stokes equations

Riccardo Milani: Wednesday 9 December at 16:00 via this link with meeting ID: 996 3774 0482 and code: 190673 We develop face-based Compatible Discrete Operator (CDO-Fb) schemes for the unsteady, incompressible Stokes and Navier–Stokes equations. We introduce operators discretizing the gradient, the divergence, and the convection term. It is proved that the discrete divergence operator allows one to recover a discrete inf-sup condition. Moreover, the discrete convection operator is dissipative, a paramount property for the energy balance. The scheme is first tested in the steady case on general and deformed meshes in order to highlight the flexibility and the robustness of the CDO-Fb discretization. The focus is then moved onto the time-stepping techniques. In particular, we analyze the classical monolithic approach, consisting in solving saddle-point problems, and the Artificial Compressibility (AC) method, which allows one to avoid such saddle-point systems at the cost of relaxing the mass balance. Three classic techniques for the treatment of the convection term are investigated: Picard iterations, the linearized convection and the explicit convection. Numerical results stemming from first-order and then from second-order time-schemes show that the AC method is an accurate and efficient alternative to the classical monolithic approach.

## November 26 – Koondanibha Mitra: A posteriori error bounds for the Richards equation

Koondanibha Mitra: Thursday 26 November at 16:00 via this link Richards equation is commonly used to model the flow of water and air through the soil, and serves as a gateway equation for multiphase flow through porous domains. It is a nonlinear advection-reaction-diffusion equation that exhibits both elliptic-parabolic and hyperbolic-parabolic kind of degeneracies. In this study, we provide fully computable, locally space-time efficient, and reliable a posteriori error bounds for numerical solutions of the fully degenerate Richards equation. This is achieved in a variation of the $H^1(H^{-1})\cap L^2(L^2) \cap L^2(H^1)$ norm characterized by the minimum regularity inherited by the exact solutions. For showing global reliability, a non-local in time error estimate is derived individually for the $H^1(H^{-1})$, $L^2(L^2)$ and the $L^2(H^1)$ error components with a maximum principle and a degeneracy estimator being used for the last one. Local and global space-time efficient error bounds are obtained, and error contributors such as flux and time non-conformity, quadrature, linearisation, data oscillation, are identified and separated. The estimates hold also in space-time adaptive settings. The predictions are verified numerically and it is shown that the estimators correctly identify the errors up to a factor in the order of unity.