February 7 – Gregor Gantner: Optimal adaptivity for isogeometric finite and boundary element methods

Gregor Gantner: Thursday 7 February at 11 am, A415 Inria Paris. The CAD standard for spline representation in 2D or 3D relies on tensor-product splines. To allow for adaptive refinement, several extensions have emerged, e.g., analysis-suitable T-splines, hierarchical splines, or LR-splines. All these concepts have been studied via numerical experiments, but there exists only little literature concerning the thorough analysis of adaptive isogeometric methods. The work [1] investigates linear convergence of the weighted-residual error estimator (or equivalently: energy error plus data oscillations) of an isogeometric finite element method (IGAFEM) with truncated hierarchical B-splines. Optimal convergence was indepen- dently proved in [2, 3]. In [3], we employ hierarchical B-splines and propose a refinement strategy to generate a sequence of refined meshes and corresponding discrete solutions. Usually, CAD provides only a parametrization of the boundary ∂Ω instead of the domain Ω itself. The boundary element method, which we consider in the second part of the talk, circumvents this difficulty by working only on the CAD provided boundary mesh. In 2D, our adaptive algorithm steers the mesh-refinement and the local smoothness of the ansatz functions. We proved linear convergence of the employed weighted-residual estimator at optimal algebraic rate in [5, 6]. In 3D, we consider an adaptive IGABEM with hierarchical splines and prove linear convergence of the estimator at optimal rate; see [6]. REFERENCES [1] A. Buffa and C. Giannelli, Adaptive isogeometric methods with hierarchical splines: error estimator and convergence. Math. Mod. Meth. Appl. S., Vol. 26, 2016. [2] A. Buffa and C. Giannelli, Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates. Math. Mod. Meth. in Appl. S., Vol. 27, 2017. [3] G. Gantner, D. Haberlik, and Dirk Praetorius, Adaptive IGAFEM with optimal con- vergence rates: Hierarchical B-splines. Math. Mod. Meth. in Appl. S., Vol. 27, 2017. [4] G. Gantner, Adaptive…

Lire la suite

Wednesday 5 – Amina Benaceur: Model reduction for nonlinear thermics and mechanics

Amina Benaceur: Wednesday 5 December at 11:00 am, A315 Inria Paris. This thesis introduces three new developments of the reduced basis method (RB) and the empirical interpolation method (EIM) for nonlinear problems. The first contribution is a new methodology, the Progressive RB-EIM (PREIM) which aims at reducing the cost of the phase during which the reduced model is constructed without compromising the accuracy of the final RB approximation. The idea is to gradually enrich the EIM approximation and the RB space, in contrast to the standard approach where both constructions are separate. The second contribution is related to the RB for variational inequalities with nonlinear constraints. We employ an RB-EIM combination to treat the nonlinear constraint. Also, we build a reduced basis for the Lagrange multipliers via a hierarchical algorithm that preserves the non-negativity of the basis vectors. We apply this strategy to elastic frictionless contact for non-matching meshes. Finally, the third contribution focuses on model reduction with data assimilation. A dedicated method has been introduced in the literature so as to combine numerical models with experimental measurements. We extend the method to a time-dependent framework using a POD-greedy algorithm in order to build accurate reduced spaces for all the time steps. Besides, we devise a new algorithm that produces better reduced spaces while minimizing the number of measurements required for the final reduced problem.

Lire la suite

January 9 – Zhaonan Dong: hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes

Zhaonan Dong: Wednesday 9 January at 11 am, A415 Inria Paris. PDE models are often characterised by local features such as solution singularities/layers and domains with complicated boundaries. These special features make the design of accurate numerical solutions challenging, or require huge amount of computational resources. One way of achieving complexity reduction of the numerical solution for such PDE models is to design novel numerical methods which support general meshes consisting of polygonal/polyhedral elements, such that local features of the model can be resolved in efficiently by adaptive choices of such general meshes. In this talk, we will review the recently developed hp-version symmetric interior penalty discontinuous Galerkin (dG) finite element method for the numerical approximation of PDEs on general computational meshes consisting of polygonal/polyhedral (polytopic) elements. The key feature of the proposed dG method is that the stability and hp-version a-priori error bound are derived based on the specific choice of the interior penalty parameters which allows for edges/faces degeneration. Moreover, under certain practical mesh assumptions, the proposed dG method was proven to be available to incorporate very general polygonal/polyhedral elements with an arbitrary number of faces. Because of utilising general shaped elements, the dG method shows a great flexibility in designing an adaptive algorithm by refining or coarsening general polytopic elements. Especially for solving the convection-dominated problems on which boundary and interior layers may appear and need a lot of degrees of freedom to resolve. Finally, we will present several numerical examples through different classes of linear PDEs to highlight the practical performance of the proposed method.

Lire la suite

Journée contrat cadre IFPEN / Inria

26 novembre 2018 Salle Jacques-Louis Lions 2 Bâtiment C, rez-de-chaussée Inria, 2 rue Simone Iff, Paris 12ème Comment venir Programme 9h45h-10h15 : Accueil autour d’un café 10h15-10h30 : Avenant contrat cadre IFPEN/INRIA (2018-2022), ajout de la nouvelle thématique « Intelligence Artificielle et science des données » (Van Bui-Tran) 10h30-11h : Méthodes de décomposition de domaine via le langage dédié FreeFem++ (Fréderic Nataf) 11h-11h30 : Goal-oriented a posteriori error estimation for conforming and nonconforming approximations with inexact solvers (Soleiman Yousef) 11h30-12h : Reduced-Basis method for two-phase darcean flows (Sébastien Boyaval) 12h-12h30 : Etude et simulation d’un modèle stratigraphique advecto-diffusif non-linéaire avec frontières mobiles (Huy Quang Tran) 12h30-14h00 : Repas: Crêperie Paris Breizh, 177 avenue Daumesnil, 75012 Paris 14h00-14h30 : A posteriori error estimates and adaptive stopping criteria for a compositional two-phase flow with nonlinear complementarity constraints (Jad Dabaghi) 14h30-15h00 : Adaptive resolution of linear systems based on error estimators (Zakariae Jorti) 15h00-15h30 : Modèle cinétique pour le transport réactif (Bastien Hamlat) 15h30-15h45 : Pause-café 15h45-16h15 : Stratégie(s) pour améliorer la robustesse de l’algorithme de Newton intervenant dans la résolution de l’équation de Richards (Guillaume Enchéry) 16h15-16h20 : Conclusion de la journée (Martin Vohralík)

Lire la suite

December 13 – Maxime Breden: An introduction to a posteriori validation techniques, illustrated on the Navier-Stokes equations

Maxime Breden: Thursday 13 December at 11 am, A415 Inria Paris. The aim of a posteriori validation techniques is to obtain mathematically rigorous and quantitative existence theorems, using numerical simulations. Given an approximate solution, the general strategy is to combine a posteriori estimates with analytical ones to apply a fixed point theorem, which then yields the existence of a true solution in an explicit neighborhood of the approximate one. In the first part of the talk, I’ll present the main ideas in more detail, and describe the general framework in which they are applicable. In the second part, I’ll then focus on a specific example and explain how to validate a posteriori periodic solutions of the Navier-Stokes equations with a Taylor-Green type of forcing. This is joint work with Jan Bouwe van den Berg, Jean-Philippe Lessard and Lennaert van Veen.

Lire la suite

Axioms of adaptivity — mini-course by C. Carstensen

Tuesday 11 September — Thursday 13 September at 9am (three 2-hour sessions), A415 Inria Paris Files for the participants (available only during the course): AoA1 AoA2 AoA3 Mini-course by C. Carstensen (Humboldt-Universität zu Berlin, Germany): The lecture series on the optimal rates of adaptive mesh-refining algorithms in computational PDEs provides an introduction to the mathematics of optimal rates based on the standard Dörfler marking in a collective refinement strategy. The focus is on the thorough insight into the mathematics for the simplest meaningful setting with elementary tools like the trace inequality, inverse estimate, plus several forms of triangle and Cauchy inequalities. Solely four axioms guarantee the optimality in terms of the error estimators outlined in Part 1 of the lectures. This general framework covers a huge part of the existing literature on optimal rates of adaptive schemes and is exemplified for the 2D Poisson model problem on polygonal domains. Part 2 gives the outline of the proof of optimal rates with linear convergence and a comparison lemma as Stevenson’s key argument for the optimality. The abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method. The local discrete efficiency of the error estimator is neither needed to prove convergence nor utilised for the quasi-optimal convergence behaviour of the error estimator. Efficiency exclusively characterises the approximation classes involved in terms of the best-approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on any efficiency constant. Some general quasi-Galerkin orthogonality is not only sufficient, but also necessary for the R-linear convergence of the error estimator. Part 3 discusses the lowest-order conforming finite element method based on triangles and provides proofs of the stability, reduction, and discrete localised reliability of the explicit residual-based…

Lire la suite

April 16 – Simon Lemaire: An optimization-based method for the numerical approximation of sign-changing PDEs

Simon Lemaire: Thursday 16 April at 3 pm, A415 Inria Paris. We are interested in physical settings presenting an interface between a classical (positive) material and a (negative) metamaterial, in such a way that the coefficients of the model change sign in the domain. We study, in the « elliptic » case, the numerical approximation of such sign-shifting problems. We introduce a new numerical method, based on domain decomposition and optimization, that we prove to be convergent, as soon as, for a given right-hand side, the problem admits a solution that is unique. The proof of convergence does not rely on any symmetry assumption on the mesh family with respect to the sign-changing interface. In that respect, it gives a more convenient alternative to T-coercivity based approximation in the situations when the latter is applicable, whereas it constitutes a new paradigm in the situations when the latter is not. We illustrate our findings on a comprehensive set of test-cases.

Lire la suite

February 20 – Thirupathi Gudi: An energy space based approach for the finite element approximation of the Dirichlet boundary control problem.

Thirupathi Gudi: Tuesday 20 February at 3 pm, A415 Inria Paris. In this talk, we review some approaches for formulating the Dirichlet boundary control problem and then we present a new energy space based approach. We show that this new approach allows high regularity for both optimal control and the optimal state. Using, the optimality conditions at continuous level, we propose a finite element method for numerical solution and derive subsequent error estimates. We show some numerical experiments to illustrate the method.

Lire la suite

February 15 – Franz Chouly: About some a posteriori error estimates for small strain elasticity

Franz Chouly: Thursday 15 February at 2 pm, A415 Inria Paris. In the first part of this talk, we will present a residual based a posteriori error estimate for contact problems in small strain elasticity, discretized with finite elements and Nitsche’s method. Upper and lower bounds are established under a saturation assumption. This theoretical results will be illustrated by some numerical experiments (joint work with Mathieu Fabre, Patrick Hild, Jérôme Pousin and Yves Renard). In the second part of this talk, we will present preliminary results on goal oriented error estimates for soft-tissue biomechanics, still under small strain assumptions. The performance of the Dual Weighted Residual method will be assessed for two simplified scenarios involving tongue muscular activation, and contraction of the arterial wall. Open mathematical questions and the potential interest of such a methodology for computational biomechanics will be discussed (joint work with Stéphane Bordas, Marek Bucki, Michel Duprez, Vanessa Lleras, Claudio Lobos, Alexei Lozinski, Pierre-Yves Rohan and Satyendra Tomar).

Lire la suite