# Internal Seminar

## 2022 – 2023

• 28 November 2022 – 11h00 to 12h00 Xuefeng LiuGuaranteed eigenvalue/eigenfunction computation and its application to shape optimization problems. (abstract)
• 17 November 2022 – 11h00 to 12h00 Fabio ViciniFlow simulations on porous fractured media: a small numerical overview from my perspective. (abstract)
• 20 October 2022 – 10h00 to 11h00 Iuliu Sorin PopNon-equilibrium models for flow in porous media. (abstract)
• 06 October 2022 – 15h00 to 16h00 Rekha KhotNonconforming virtual elements for the biharmonic equation with Morley degrees of freedom on polygonal meshes. (abstract)

## 2021 – 2022

• 21 September 2022 – 15h00 to 16h00 Alexandre IMPERIALENumerical methods for time domain wave propagation problems applied to ultrasonic testing modelling. (abstract)
• 16 June 2022 – 11h30 to 12h30 Cherif AmroucheElliptic Problems in Lipschitz and in $C^{1,1}$ Domains. (abstract)
• 13 June 2022 – 11h00 to 12h00 Jean-Luc GuermondInvariant-domain preserving IMEX time stepping methods. (abstract)
• 5 May 2022 – 11h00 to 12h00 Daniel Zegarra VasquezSimulation d’écoulements monophasiques en milieux poreux fracturés par la méthode des éléments finis mixtes hybrides. (abstract)
• 19 April 2022 – 14h00 to 15h00 Christos XenophontosFinite Element approximation of singularly perturbed eigenvalue problems. (abstract)
• 14 April 2022 – 11h00 to 12h00: Idrissa NiakhStable model reduction for linear variational inequalities with parameter-dependent constraints. (abstract)
• 7 April 2022 – 17h00 to 18h00: Christoph LehrenfeldEmbedded Trefftz Discontinuous Galerkin methods. (abstract)
• 24 March 2022 – 11h00 to 12h00: Miloslav Vlasak: A posteriori error estimates for discontinuous Galerkin method. (abstract)
• 10 March 2022 – 11h00 to 12h00: Ruma Maity: Parameter dependent finite element analysis for ferronematics solutions. (abstract)
• 3 February 2022 – 11h00 to 12h00: Pierre Matalon: An h-multigrid method for Hybrid High-Order discretizations of elliptic equations. (abstract)
• 27 January 2022 – 11h00 to 12h00: Frédéric LebonOn the modeling of nonlinear imperfect solid/solid interfaces by asymptotic techniques. (abstract)
• 20 January 2022 – 11h00 to 12h00: Isabelle RamièreAutomatic multigrid adaptive mesh refinement with controlled accuracy for quasi-static nonlinear solid mechanics. (abstract)
• 13 January 2022 – 11h00 to 12h00: Koondanibha Mitra: A posteriori estimates for nonlinear degenerate parabolic and elliptic equations. (abstract)
• 10 December 2021 – 11h00 to 12h00: Gregor GantnerApplications of a space-time first-order system least-squares formulation for parabolic PDEs. (abstract)
• 25 November 2021 – 11h00 to 12h00: Pierre GosseletAsynchronous Global/Local coupling. (abstract)
• 24 November 2021 – 10h30 to 11h30: Grégory EtangsaleA primal hybridizable discontinuous Galerkin method for modelling flows in fractured porous media. (abstract)

## 2020 – 2021

• 06 September 2021 – 15h00 to 16h00: Rolf Stenberg: Nitsche’s Method for Elastic Contact Problems. (abstract)
• 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

## 17th November – Fabio Vicini: Flow simulations on porous fractured media: a small numerical overview from my perspective

Fabio Vicini Thursday 17th Nov at 11:00   ABSTRACT: The presentation will focus on some numerical aspects of the simulations of porous-fractured media I have experienced so far. Flow simulations on porous fractured media contain many numerical challenging issues related to complex geometries and the severe size of the computational domain. In real applications, indeed, the network of fractures immersed in the rock matrix is usually generated according to known probabilistic distributions, which lead to a huge number of fractures with a multi-scale distribution in sizes. For these reasons, tailored numerical methods are s to overcome these problems and for efficient handling of the computational resources. I will show two different approaches to tackle the classic Darcy single-phase problem with the Discrete Fracture Networks (DFNs) model: one related to a constrained optimization problem and the second related to the modern Virtual Element (VEM) framework. Moreover, I will present the possibility of investigating the Darcy problem on DFNs as a parameterized differential problem with a Reduced Basis (RB) technique combined with a-posteriori error control. Finally, I will discuss the possibility of extending the VEM approach for the simulation of a two-phase flow of immiscible fluids.

## 28th November – Xuefeng Liu: Guaranteed eigenfunction computation and its application to shape optimization problems

Xuefeng Liu Monday 28th Nov at 11:00   ABSTRACT: We are concerned with the guaranteed computation of the eigenfunction (or eigenspace) of differential operators. Upon the setting of eigenvalue problems, three algorithms are proposed to give rigorous and efficient error estimation for the approximate eigenfunctions: Algorithm I: Rayleigh quotient error based algorithm; Algorithm II: Variational residual error based algorithm; Algorithm III: Projection error based algorithm. The guaranteed eigenfunction computation is applied to solving shape optimization problems. For example, the minimization of the Laplacian eigenvalue over polygonal domains. By explicitly evaluating the Hadamard shape derivative with guaranteed computation of both eigenvalues and eigenfunctions, we provide a computer-assisted proof to declare that the equilateral triangle minimizes the first Laplacian under certain non-homogeneous Neumann boundary condition under the radius constraint condition. References: 1. R. Endo and X. Liu, Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error constant estimation, https://arxiv.org/abs/2209.13415. 2. X. Liu and T. Vejchodsky, Projection error-based guaranteed L2 error bounds for finite element approximations of Laplace eigenfunctions, https://arxiv.org/abs/2211.03218. 3.X. Liu and T. Vejchodsky, Fully computable a posteriori error bounds for eigenfunctions, Numer. Math. 152 (2022), 183–221.

## 20th October – Iuliu Sorin Pop: Non-equilibrium models for flow in porous media

Iuliu Sorin Pop Thursday 20th Oct at 10:00   ABSTRACT: We discuss various aspects related to non-equilibrium mathematical models for porous media flow. Typically, such models assume that quantities like saturation, phase pressure differences, or relative permeability are related by monotone, algebraic relationships. Under such assumptions, the solutions satisfy the maximum principle. On the other hand, experimental work published in the past decades report that phenomena like saturation overshoot, or the formation of finger profiles have been observed whenever the flow is sufficiently rapid. Such results are ruled out by standard, equilibrium models. This is the main motivation to consider non-equilibrium models, where dynamic or hysteretic effects are included in the above-mentioned relationships. The resulting models are nonlinear evolution systems of (pseudo-)parabolic and possibly degenerate equations, and involving differential inclusions. For such problems, we present first some results concerning the derivation of such models from the pore scale to the Darcy scale, the existence and uniqueness of weak solutions, and discuss different numerical schemes. This includes aspects like the rigorous convergence of the discretization, and solving the emerging nonlinear time-discrete or fully discrete problems. This is joint work with X. Cao (Toronto), S. Karpinski (Munich), S. Lunowa (Munich), K. Mitra (Hasselt), F.A. Radu (Bergen)

## 06th October – Rekha Khot: Nonconforming virtual elements for the biharmonic equation with Morley degrees of freedom on polygonal meshes

Rekha Khot Thursday 06th Oct at 15:00   ABSTRACT: The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution $u \in V := H_0^2(Ω)$ to the biharmonic equation. Two nonconforming virtual element spaces have been introduced in [1, 4] for $H^3$ regular solutions, while a medius analysis in [3] allows minimal regularity. In this talk, we will discuss an abstract framework (cf. [2]) with two hypotheses (H1)-(H2) for unified stability and a priori error analysis of at least two different discrete spaces (even a mixture of those). A smoother J allows rough source terms $F \in V^∗ = H^{−2}(\Omega)$. The a priori and a posteriori error analysis circumvents any trace of second derivatives by a computable conforming companion operator $J : V_h \rightarrow V$ from the nonconforming virtual element space $V_h$. The operator J is a right-inverse of the interpolation operator and leads to optimal error estimates in piecewise Sobolev norms without any additional regularity assumptions on $$u \in V$$. As a smoother the companion operator modifies the discrete right-hand side and then allows a quasi-best approximation. An explicit residual-based a posteriori error estimator is reliable and efficient up to data oscillations. Numerical examples display the predicted empirical convergence rates for uniform and optimal convergence rates for adaptive mesh-refinement. [1] P. F. Antonietti, G. Manzini, and M. Verani, The fully nonconforming virtual element method for biharmonic problems, Math. Models Methods Appl. Sci., 28 (2018), pp. 387–407. [2] C. Carstensen, R. Khot, and A. K. Pani, Nonconforming virtual elements for the biharmonic equation with Morley degrees of freedom on polygonal meshes, arXiv:2205.08764, (2022). [3] J. Huang and Y. Yu, A medius error analysis for nonconforming virtual element methods for Poisson and biharmonic equations, J. Comput. Appl. Math., 386 (2021), p. 21. [4] J. Zhao, B. Zhang,…

## 21th September – Alexandre IMPERIALE: Numerical methods for time domain wave propagation problems applied to ultrasonic testing modelling

Alexandre IMPERIALE Wednesday 21th Sep at 15:00   ABSTRACT: The Department of Imaging and Simulation for Control (DISC) at CEA – LIST is dedicated to developing Non-Destructive Testing (NDT) methods in advanced industrial contexts (e.g. nuclear energy, aeronautic, or petrochemical industries). Amongst different testing modalities, Ultrasonic Testing (UT) is widespread and plays a major role in numerous industrial processes. It relies on the propagation of mechanical waves in order to probe critical specimens, often formed with complex media – in terms of geometry or material properties. To fully understand the capabilities and performances of UT experiments, the DISC is developing modelling tools for wave propagation, which are bound to incorporate the CIVA platform [1] – a commercial software dedicated to NDT modelling. When commercializing modelling tools, efficiency and robustness are key features. This talk focuses on numerical methods for wave propagation problems that address these challenges. In particular, we show how the combination of Spectral Finite Elements [2] with the Mortar Element method [3] – a domain decomposition approach – can be tuned to render efficient and robust algorithms [4]. Coupled with high-frequency asymptotic methods [5] (e.g. ray-based algorithms), this combination forms the basis of various advances in the CIVA platform. This approach is typically used to modelling UT experiments relying on bulk wave propagation, e.g. for testing welds (typical nuclear energy industry applications) or composite materials [6] (typical aeronautic industry applications). Additionally, we present some details on ongoing works related to other UT configurations, such as numerical modelling of wave propagation through thin layers – based on effective transmission conditions – or within plate-like geometries – achieved through dedicated time schemes. [1] http://www.extende.com/civa-in-a-few-words [2] Cohen, G. C. “Higher-order numerical methods for transient wave equations”. Berlin: Springer, 2002 Vol. 5. [3] F. Ben Belgacem, Y. Maday, “The mortar element method…

## 13th June – Jean-Luc Guermond: Invariant-domain preserving IMEX time stepping methods

Jean-Luc Guermond Thursday 13th June at 11:00   ABSTRACT: I will present high-order time discretizations of a Cauchy problem where the evolution operator comprises a hyperbolic part and a parabolic part (say diffusion and stiff relaxation terms). The said problem is assumed to possess an invariant domain. I will propose a technique that makes every implicit-explicit (IMEX) time stepping scheme invariant domain preserving and mass conservative. The IMEX scheme is written in incremental form and, at each stage of the scheme, we first compute low-order hyperbolic and parabolic updates, followed by their high-order counterparts. The proposed technique, which is agnostic to the space discretization, allows to optimize the time step restrictions induced by the hyperbolic sub-step. To illustrate the proposed methodology, we derive three novel IMEX schemes with optimal efficiency and for which the implicit scheme is singly-diagonal and L-stable: a third-order, four-stage scheme; and two fourth-order schemes, one with five stages and one with six stages. The novel IMEX schemes are evaluated numerically on a stiff ODE system. We also apply these schemes to nonlinear convection-diffusion problems with stiff reaction and to compressible viscous flows possibly including grey radiation.

## 16th June – ‪Chérif Amrouche: Elliptic Problems in Lipschitz and in $C^{1,1}$ Domains

Chérif Amrouche Thursday 16th June at 11:30   ABSTRACT: We are interested here in questions related to the maximal regularity of solutions to elliptic problems with Dirichlet or Neumann boundary conditions (see ([1]). For the last 40 years, many works have been concerned with questions when Ω is a Lipschitz domain. Some of them contain incorrect results that are corrected in the present work. We give here new proofs and some complements for the case of the Laplacian (see [3]), the Bilaplacian ([2] and [6]) and the operator div (A∇) (see ([5]) when A is a matrix or a function. And we extend this study to obtain other regularity results for domains having an adequate regularity. We give also new results for the Dirichlet-to-Neumann operator for Laplacian and Bilaplacian. Using the duality method, we can then revisit the work of Lions-Magenes [4], concerning the so-called very weak solutions, when the data are less regular. References : [1]  C. Amrouche and M. Moussaoui. Laplace equation in smooth or non smooth do- mains. Work in Progress. [2]  B.E.J. Dahlberg, C.E. Kenig, J. Pipher and G.C. Verchota. Area integral estimates for higher-order elliptic equations and systems. Ann. Inst. Fourier, 47-5, 1425– 1461, (1997). [3]  D. Jerison and C.E. Kenig. The Inhomogeneous Dirichlet Problem in Lipschitz Domains, J. Funct. Anal. 130, 161–219, (1995). [4]  J.L. Lions and E. Magenes. Problemes aux limites non-homogenes et applications, Vol. 1, Dunod, Paris, (1969). [5]  J. Necas. Direct methods in the theory of elliptic equations. Springer Monographs in Mathematics. Springer, Heidelberg, (2012). [6]  G.C. Verchota. The biharmonic Neumann problem in Lipschitz domains. Acta Math. 194-2, 217–279, (2005).

## 5th May – ‪Daniel Zegarra Vasquez: Simulation of single-phase flows in fractured porous media using the mixed hybrid finite element method

Daniel Zegarra Vasquez Thursday 5th May at 11:00   ABSTRACT: In underground environments, fractures are very numerous and present at all scales, with very heterogeneous sizes. In particular for flows, they are preferential channels: flows are much faster there than in the neighboring rock. Indeed, the permeability of rock is generally about two orders of magnitude lower than that of fractures. This makes fractures play a vital role in a large number of industrial and environmental applications. These particularities of the fractured porous domain make the modeling and simulation of the flows passing through it a major challenge today for which it is necessary to develop dedicated, robust and efficient models and numerical methods. The most commonly used model for representing fractures is the discrete fracture network (DFN) in which fractures are represented as structures of codimension 1. The model of single-phase flows in fractured porous media is described in [5]. The particularity of the fractured porous problem, compared to the porous-only or fractured-only problem [3], is the coupling between the flow in the fractures and the flow in the rock. Due to the difficulties encountered in taking into account the geometric complexity of large fractured networks in simulations, the test cases recently proposed in the literature are mainly 2D, or 3D with a limited number (about ten) of fractures [1]. In this talk, we will present the nef-flow-fpm solver, which solves the stationary 3D fractured porous problem using the mixed hybrid finite element method. The method developed in the solver is inspired by [4]. To mesh the domain, a first simplical and conforming 2D mesh is generated for the DFN and for the boundaries of the domain, then a second simplical and conforming 3D mesh is generated from the first mesh. The solvers integrated in nef-flow-fpm are direct solvers,…

## 7th April – ‪Christoph Lehrenfeld: Embedded Trefftz Discontinuous Galerkin methods

Christoph Lehrenfeld Thursday 7th April at 11:00   ABSTRACT: Discontinuous Galerkin (DG) methods are widely used to discretize partial differential equations (PDEs) due to (a.o.) flexibility for designing robust methods and simplicity in terms of data structures. One major drawback of DG methods is, however, the increased number of (globally coupled) degrees of freedom (ndof) compared to, for instance, continuous Galerkin methods. One – by now established – remedy is the use of Hybrid DG methods. These allow reducing the globally coupled ndof essentially by introducing the concept of static condensation for DG methods. Thereby the dimensions of global linear systems that need to be solved for reduced from O(p^d) to O(p^{d−1}), where d is the space dimension and p is the polynomial degree of the finite element space. A different approach is the use of Trefftz DG methods, where a DG formulation is modified by restricting the finite element spaces to functions that element-wise solve the PDE at hand. This results in a similar reduction of globally coupled ndof, cf. Figure 1 below. However, due to several limitations, Trefftz DG methods have only been applied for special PDEs so far. On the one hand, the finite element spaces have to be specifically tailored for each PDE type, on the other hand, inhomogeneous equations and non-constant coefficients in the differential operators are difficult to deal with and rarely treated. In this talk, we introduce Embedded Trefftz DG methods which exploit the existence of an underlying standard DG formulation for an efficient and flexible implementation of Trefftz DG methods. Furthermore, we relax Trefftz DG methods in view of the constraints on the finite element space leading to weak Trefftz DG spaces. Both together allow us to remove the limitations of Trefftz DG methods in order to enable them for a large…

## 19th April – ‪Christos Xenophontos: Finite Element approximation of singularly perturbed eigenvalue problems

Christos Xenophontos Tuesday 19th April at 14:00   ABSTRACT: We consider singularly perturbed eigenvalue problems in one-dimension, and their numerical approximation by the (standard Galerkin) Finite Element Method (FEM). These are fourth order equations, where a small parameter multiplies the highest order derivative. We will present results for an $h$ version FEM with polynomials of degree $p$ on an exponentially graded (eXp) mesh, as well as an $hp$ version FEM on the so-called Spectral Boundary Layer (SBL) mesh. For both methods, robust optimal convergence is shown for the eigenvalues and associated eigenfunctions. Numerical results, illustrating the theory, will also be presented. This is joint work with H. G. Roos.