Seminar in EDP and Applied Mathematics

May 03, 2023 – 14h (Brazil)

Streaming: Youtube Channel | SEMINARIO DE EDP E MATEMATICA APLICADA

Event

About Seminars

Our Online Seminar is one of the most important events in Brazil, it has been held since August 2020, every Wednesday at 2 pm, Brasília time, with a frequency of 14 days. Two 40-minute lectures are presented in each session. Our speakers are world-renowned mathematicians from Europe, the United States and South America.

Featured Talks & Speakers

Michel Duprez

Inria, équipe MIMESIS, Université de Strasbourg,
-Icube, CNRS UMR 7357, Strasbourg, France

14:00h – 14:40h

phi-FEM, A fictitious domain method for finite
element methods on domains defined by level-sets

We present a new fictitious domain finite element method, well suited for elliptic problems posed in a domain given by a level-set function without requiring a mesh fitting the boundary. To impose the Dirichlet boundary conditions, we search the approximation

to the solution as a product of a finite element function with the given level-set function, also approximated by finite elements.The imposition of Neumann boundary conditions is less straightforward and requires the introduction of auxiliary variables nearthe boundary . Unlike other recent fictitious domain-type methods (XFEM, CutFEM), our approach does not need any non-standard numerical integration,neither on the cut mesh elements nor on the actual boundary. We shall present the proofs of optimal convergence of our methods on the example of Poisson equation using Lagrange finite elements of any order.We will also give numerical tests illustrating the optimal convergence of our methods and discuss the conditionning of resulting linear systems and the robustness with respect to the geometry. We will highlight the flexibility and efficiency of our method on elastic and dynamic problems. And more recently, we have proposed a phi-FEM formulation to solve particulate flows and Stokes equations.

Karine Beauchard

Univ Rennes, CNRS, IRMAR – UMR 6625, F-35000 Rennes, France

14:50h – 15:30h

Lie brackets and interpolation for controllability

This talk will survey old and recent results on the local controllability of control systems modeled by ODEs, focussing on results stated using Lie brackets of the vector fields defining the dynamics. We will propose a unified approach to determine and prove obstructions to local controllability. This approach relies on a recent Magnus-type representation formula of the state, a new Hall basis of the free Lie algebra over two generators and Gagliardo-Nirenberg interpolation inequalities. This approach allows to recover the known necessary conditions, but also to prove a conjecture of 1986 due to Kawski and many other new necessary conditions. Finally, we will see how these results translate for PDEs, in particular the Schrödinger equation. This is a joint work with Frederic Marbach, Jeremy Le Borgne and Mégane Bournissou.

About Organization

Juan Limaco – UFF – Brazil (Coordenador);
Mauro Rincon – UFRJ – Brazil
Max Souza – UFF – Brazil;
Sandra Malta – LNCC – Brazil
Marcelo Cavalcanti – UEM – Brazil;
Rui Almeida – UBI – Portugal
Roxana Lopez – UNMSM – Peru;
Diego Souza – U Sevilla – España;
Mauricio Sepulveda – UdeC – Chile;
Roberto Capistrano – UFPE – Brazil.