Internal Seminar

Internal Seminar Calendar

2023 – 2024

  • 17 October 2024 – 11h00 to 12h00 Gregor Gantner: Space-time FEM-BEM couplings for parabolic transmission problems. (abstract)
  • 10 September 2024 – 11h00 to 12h00 Carsten Carstensen: Adaptive computation of fourth-order problems. (abstract)
  • 25 April 2024 – 11h00 to 12h00 Martin Werner Licht: Computable reliable bounds for Poincaré–Friedrichs constants via Čech–de-Rham complexes. (abstract)
  • 4 April 2024 – 11h00 to 12h00 Roland Maier: A localized orthogonal decomposition strategy for hybrid discontinuous Galerkin methods. (abstract)
  • 2 April 2024 – 14h00 to 15h00 Andreas Rupp: Homogeneous multigrid for hybrid discretizations: application to HHO methods. (abstract)
  • 18 January 2024 – 14h00 to 15h00 Zoubida Mghazli: Modeling some biological phenomena via the porous media approach. (abstract)
  • 23 November 2023 – 11h00 to 12h00 Olivier Hénot: Computer-assisted proofs of radial solutions of elliptic systems on R^d. (abstract)
  • 16 November 2023 – 17h00 to 18h00 Maxime Theillard: A Volume-Preserving Reference Map Method for the Level Set Representation. (abstract)
  • 13 November 2023 – 11h00 to 12h00 Charles Parker: Implementing $H^2$-conforming finite elements without enforcing $C^1$-continuity. (abstract)
  • 09 November 2023 – 11h00 to 12h00 Maxime Breden: Computer-assisted proofs for nonlinear equations: how to turn a numerical simulation into a theorem. (abstract)

2022 – 2023

  • 25 May 2023 – 11h00 to 12h00 Martin Vohralík: A posteriori error estimates robust with respect to nonlinearities and final time. (abstract)
  • 11 May 2023 – 11h00 to 12h00 Konstantin Brenner: On the preconditioned Newton’s method for Richards’ equation. (abstract)
  • 4 May 2023 – 11h00 to 12h00 Ludmil Zikatanov: High order exponential fitting discretizations for convection diffusion problems. (abstract)
  • 23 March 2023 – 11h00 to 12h00 Marien Hanot: Polytopal discretization of advanced differential complexes.(abstract)
  • 9 February 2023 – 11h00 to 12h00 Roland Maier: Semi-explicit time discretization schemes for elliptic-parabolic problems. (abstract)
  • 2 February 2023 – 11h00 to 12h00 Simon Legrand: Parameter studies automation with Prune_rs. (abstract)
  • 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

June 22 – Valentine Rey: Goal-oriented error control within non-overlapping domain decomposition methods to solve elliptic problems

Valentine Rey: Thursday 22 June at 3 pm, A415 Inria Paris. Domain decomposition methods are robust and efficient methods to solve mechanical problems with several million degrees of freedom. Taking advantage of increasing performances of computers, they exploit the clusters-parallel architecture and are numerically scalable. Verification has been widely developed since 1980’s and proposes tools to estimate the distance between the unknown exact solution of continuous problem and the computed solution. Among those techniques, estimators based on error in constitutive relation provide constant-free upper bounds and are available for varied range of problems. In this talk, we present techniques for steering parallel computation by objective of accuracy on quantities of interest. It relies on a parallel error estimator that provides strict guaranteed upper bound and separates the algebraic error (due to the use of iterative solver) from the discretization error (due to the finite element method). This estimator enables to adapt the solver’s stopping criterion to the discretization, which avoids over resolution and useless iterations. In [*], the estimator is used for goal-oriented error estimation and classical bounds for quantities of interest are rewritten in order to separate sources of error. Finally, we benefit the information provided by the error estimator and the Krylov subspaces built during the resolution to set an auto-adaptive strategy (adaptive remeshing and recycling search directions). *V. Rey, P. Gosselet, C. Rey, Strict bounding of quantities of interest in computations based on domain decomposition, Computer Methods in Applied Mechanics and Engineering. 2015 Apr 15;287:212-28

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29 June – Gouranga Mallik: A priori and a posteriori error control for the von Karman equations

Gouranga Mallik: Thursday 29 June at 3pm, A415 Inria Paris. In this work we consider a priori and a posteriori error control for the nonsingular solution of von Karman plate bending problem. Conforming and nonconforming finite element methods are employed. Existence, uniqueness and error estimates for the discrete solution are presented. We discuss an abstract framework for a posteriori error control which includes conforming and nonconforming finite element methods. This allows us to compute reliable and efficient local estimators. The key ingredients in establishing well-posedness of the discrete problem rely on the linearization of the continuous problem and suitable enrichment operator. Numerical experiments are performed to justify the theoretical results.

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15 June – Patrik Daniel: An adaptive hp-refinement strategy with computable guaranteed error reduction factors

Patrik Daniel: Thursday 15 June at 3:30pm, A415 Inria Paris. We propose a new practical adaptive refinement strategy for hp-finite element approximations of elliptic problems. Following some recent theoretical developments in polynomial-degree-robust a posteriori error analysis, we solve two complementary classes of discrete local problems on the vertex-based patches. The first class involves the solution on each patch of a mixed finite element problem with homogeneous Neumann boundary conditions, which leads to an H(div,Ω)-conforming equilibrated flux. This in turns yields a guaranteed upper bound on the error and serves to mark elements for refinement via a Dörfler bulk criterion. The second class of local problems involves the solution, on each marked patch only, of two separate primal finite element problems with homogeneous Dirichlet boundary conditions, which serve to decide between h-, p-, or hp-refinement. Altogether, we show that these ingredients lead to a computable error reduction factor; we guarantee that while performing the hp-adaptive refinement as suggested, the error will be reduced at least by this factor on the next hp-mesh. In a series of numerical experiments in two space dimensions, we first study the accuracy of our predicted reduction factor: in particular, we measure the ratio of the predicted reduction factor relative to the true error reduction, and we find that it is very close to the optimal value of one for both smooth and singular exact solutions. Finally, we study the overall performance of the proposed hp-refinement strategy on some test cases, for which we observe effectivity indices very close to one and exponential convergence rates.

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15 June – Jad Dabaghi: Adaptive inexact semi-smooth Newton methods for a contact between two membranes

Jad Dabaghi: Thursday 15 June at 3pm, A415 Inria Paris. We propose an adaptive inexact version of a class of semi-smooth Newton methods. As a model problem, we consider the system of variational inequalities describing the contact between two membranes and its finite element discretization. Any iterative linearization algorithm like the Newton-min, Newton-Fisher Burmeister is taken into account, as well as any iterative linear algebraic solver. We prove an a posteriori error estimate between the exact solution and the approximate solution which is valid on any step of the linearization and algebraic resolution. This estimate is based on discretization and algebraic flux reconstructions, where the latter one is obtained on a hierarchy of nested meshes. The estimate distinguishes the discretization, linearization, and algebraic components of the error and allows us to formulate adaptive stopping criteria for both solvers. Numerical experiments for the semi-smooth Newton-min algorithm in combination with the GMRES solver confirm the efficiency of the method.

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June 1 – Martin Eigel: Adaptive stochastic FE for explicit Bayesian inversion with hierarchical tensor representations

Martin Eigel: Thursday 1 June at 10:45 am, A415 Inria Paris. We consider a class of linear PDEs with stochastic coefficients which depend on a countable (inifinite) number of random parameters. As an alternative to classical Monte Carlo sampling techniques, a functional discretisation of the stochastic space in generalised polynomial chaos may lead to significantly improved (optimal) convergence rates. However, when employed in the context of Galerkin methods, the arising algebraic systems are very high-dimensional and quickly become intractable to computations. As a matter of fact, this is an exemplary example for the curse of dimensionality with exponential growth of complexity which makes model reduction techniques inevitable. In the first part, we discuss two approaches for this: (1) a posteriori adaptivity and exploitation of sparsity of the solution, and (2) low-rank compression in a hierarchical tensor format. In the second part, the low-rank discretisation is used as an efficient representation of the stochastic model for Bayesian inversion. This is an important application in Uncertainty Quantification where one is interested in determining the (statistics of) parameters of the model based on a set of noisy measurements. In contrast to popular sampling techniques such as MCMC, we derive an explicit representation of the posterior densities. The examined sampling-free Bayesian inversion is adaptive in all discretisation parameters. Moreover, convergence of the method is shown.

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June 1 – Joscha Gedicke: An adaptive finite element method for two-dimensional Maxwell’s equations

Joscha Gedicke: Thursday 1 June at 10 am, A415 Inria Paris. We extend the Hodge decomposition approach for the cavity problem of two-dimensional time harmonic Maxwell’s equations to include the impedance boundary condition, with anisotropic electric permittivity and sign changing magnetic permeability. We derive error estimates for a P_1 finite element method based on the Hodge decomposition approach and develop a residual type a posteriori error estimator. We show that adaptive mesh refinement leads empirically to smaller errors than uniform mesh refinement for numerical experiments that involve metamaterials and electromagnetic cloaking. The well-posedness of the cavity problem when both electric permittivity and magnetic permeability can change sign is also discussed and verified for the numerical approximation of a flat lens experiment.

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June 1st 2017 – Quang Duc Bui – Coupled Parareal-Schwarz Waveform relaxation method for advection reaction diffusion equation in one dimension

Quang Duc Bui: Thursday 1 June at 11:30am, A415 Inria Paris. Parareal method is a numerical method to solve time-evolution problems in parallel, which uses two propagators: the coarse – fast and inaccurate – and the fine – slow but more accurate. Instead of running the fine propagator on the whole time interval, we divide the time space into small time intervals, where we can run the fine propagator in parallel to obtain the desired solution, with the help of the coarse propagator and through parareal steps. Furthermore, each local subproblem can be solved by an iterative method, and instead of doing this local iterative method until convergence, one may perform only a few iterations of it, during parareal iterations. Propagators then become much cheaper but sharply lose their accuracy, and we hope that the convergence will be achieved across parareal iterations. Here, we propose to couple Parareal with a well-known iterative method – Schwarz Waveform Relaxation (SWR)- with only few SWR iterations in the fine propagator and with a simple coarse propagator deduced from Backward Euler method. We present the analysis of this coupled method for 1-dimensional advection reaction diffusion equation, for this case the convergence is at least linear. We also give some numerical illustrations for 1D and 2D parabolic equations, which shows that the convergence is much faster in practice.

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May 16 – Quanling Deng: Dispersion Optimized Quadratures for Isogeometric Analysis

Quanling Deng: Tuesday 16 May at 3 pm, A415 Inria Paris. The isogeometric analysis (IgA) is a powerful numerical tool that unifies the finite element analysis (FEA) and computer-aided design (CAD). Under the framework of FEA, IgA uses as basis functions those employed in CAD, which are capable of exactly represent various complex geometries. These basis functions are called the B-Splines or more generally the Non-Uniform Rational B-Splines (NURBS) and they lead to an approximation which may have global continuity of order up to $p-1$, where $p$ is the order of the underlying polynomial, which in return delivers more robustness and higher accuracy than that of finite elements. We apply IgA to wave propagation and structural vibration problems to study their dispersion and spectrum properties. The dispersion and spectrum analysis are unified in the form of a Taylor expansion for eigenvalue errors. By blending optimally two standard Gaussian quadrature schemes for the integrals corresponding to the stiffness and mass, the dispersion error of IgA is minimized. The blending schemes yield two extra orders of convergence (superconvergence) in the eigenvalue errors, while the eigenfunction errors are of optimal convergence order. To analyze the eigenvalue and eigenfunction errors, the Pythagorean eigenvalue theorem (Strang and Fix, 1973) is generalized to establish an equality among the eigenvalue, eigenfunction (in L2 and energy norms), and quadrature errors.

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May 11 – Sarah Ali Hassan: A posteriori error estimates and stopping criteria for solvers using domain decomposition methods and with local time stepping

Sarah Ali Hassan: Thursday 11 May at 3 pm, A415 Inria Paris. In this work we develop a posteriori error estimates and stopping criteria for domain decomposition (DD) methods with optimized Robin transmission conditions on the interface. Steady diffusion equation using the mixed finite element (MFE) discretization as well as in the heat equation using the MFE method in space and the discontinuous Galerkin scheme in time are analysed. For the heat equation, a global-in-time domain decomposition method is used, allowing for different time steps in different subdomains. We bound the error between the exact solution of the PDE and the approximate numerical solution at each iteration of the domain decomposition algorithm. Different error components (domain decomposition, space discretization, time discretization) are distinguished, which allows us to define efficient stopping criteria for the DD algorithm. The estimates are based on the reconstruction techniques for pressures and fluxes. Numerical experiments illustrate the theoretical findings.

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June 6 – Ivan Yotov: Coupled multipoint flux and multipoint stress mixed finite element methods for poroelasticity

Ivan Yotov: Tuesday 6 June at 11 am, A415 Inria Paris. We discuss mixed finite element approximations for the Biot system of poroelasticity. We employ a weak stress symmetry elasticity formulation with three fields – stress, displacement, and rotation, as well as a mixed velocity-pressure Darcy formulation. The method is reduced to a cell-centered scheme for the displacement and the pressure, using the multipoint flux mixed finite element method for flow and the recently developed multipoint stress mixed finite element method for elasticity. The methods utilize the Brezzi-Douglas-Marini spaces for velocity and stress and a trapezoidal-type quadrature rule for integrals involving velocity, stress, and rotation, which allows for local flux, stress, and rotation elimination. We perform stability and error analysis and present numerical experiments illustrating the convergence of the method and its performance for modeling flows in deformable reservoirs. This is joint work with Ilona Ambartsumyan and Eldar Khattatov, University of Pittsburgh.

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