W o r k s h o p M T M 2 0 0 8 - 0 3 5 4 1 P r o j e c t

B C A M H e a d q u a r t e r s - B i z k a i a Te c h n o l o g y P a r k , B . 5 0 0J u n e , 3 0 t h - J u l y, 1 s t , 2 0 0 9

1

MTM2008-03541 basic data

2

BASIC DATA:

Project financed by: MICINN through the VI I+D+i national plan,

Title: Partial Differential Equations: Analysis, Control, Numerics and Applications

Reference: MTM2008-03541

Principal Investigator: ENRIQUE ZUAZUA

Research Center: IKERBASQUE / BCAM

Duration: From January 1st, 2009 to December 31st, 2011

Website: http://www.bcamath.org/projects/MTM2008-03541

3

Program schedule

4

5

June, 30th ! ! 10:50 - 11:00! ! ! Presentation of BCAM and MTM2008 Project

Enrique ZUAZUA - zuazua@bcamath.org

Presentation of BCAM - Basque Center for Applied Mathematics (Basque Country) and MTM2008-03541 project

6

June, 30th ! ! 11:00 - 11:15! ! ! Turbulence models for fluids and optimization

Carlos CASTRO - carlos.castro@upm.es

Abstract:

We present the main turbulence models for fluids and the basic numerical techniques used for. We also present the adjoint calculus for these models and some of the mathematical problems encountered in the optimization.

! ! !! ! ! 11:15 - 11:30! ! ! Questions and comments

7

June, 30th ! ! 11:30 - 12:00! ! ! COFFEE BREAK

BCAM Headquarters

8

June, 30th ! ! 12:00 - 12:15! ! ! Nuevos desarrollos y líneas de investigación en el ámbito del adjunto continuo aplicado al diseño aeronáutico

Francisco PALACIOS - francisco.palacios@upm.es

Abstract:

El objetivo de la charla será resumir el trabajo desarrollado entorno a la aplicación en el ámbito industrial de técnicas inspiradas en la teoría de control para el diseño aerodinámico. Fundamentalmente se abordarán tres aspectos: implementación eficiente del método adjunto continuo, diseño en configuraciones transónicas y diseño en sistemas gobernados por las ecuaciones RANS.

! ! ! 12:15 - 12:30! ! ! Questions and comments

9

June, 30th ! ! 12:30 - 12:45! ! ! Concentrated solutions for the finite-differences and discontinuous Galerkin semi-discretizations of the 1-d wave equation

Aurora MARICA - marica@bcamath.org

Abstract:

We will present some techniques of constructing concentrated solutions for some space semi-discretizations of the 1-d wave equation using the classical finite-differences scheme and a discontinuous Galerkin method, the so-called Symmetric Interior Penalty discontinuous Galerkin method.The existence of such concentrated solutions proves in particular the lack of uniform observability inequalties when concentration takes place around wave numbers where the corresponding group velocity vanishes. The main idea to obtain concentrated solutions is to consider initial data whose Fourier transform is supported in a small neighborhood of the wave number where the group velocity vanishes. The observability constant is proved to increse polynomially with an order depending on the regularity of the Fourier transform of the initial data.! ! ! 12:45 - 13:00! ! ! Questions and comments

10

June, 30th ! ! 13:00 - 15:00! ! ! LUNCH

BCAM Headquarters

11

June, 30th ! ! 15:00 - 15:15! ! ! The problem of maximizing the minimal bisecting chord: !a solution for Zindler sets

Aldo PRATELLI - pratelli@unipv.it

Abstract:

Let us consider a convex planar set E: among all the chords bisecting the area, there is a minimal one. An old problem (presented also in the book "Unsolved problems in Geometry", 1991, and known as "Santalo' problem") is the following: if we fix this minimal length, which set has the smallest possible area? It is reasonable to guess that this set must be the ball, but thanks to the work of Zindler (1921) it is known that it is not so. In particular, Zindler shows that there are sets which have all the bisecting chords of the same length, and which are smaller than the ball: these sets are often referred to as Zindler sets. A big work has been done in last decades to study the properties of the convex sets with respect to the minimal bisecting length, in particular in the case of Zindler sets, but the problem of Santalo' is still open. We give the answer of this problem in the class of Zindler sets.

12

! ! !! ! ! 15:15 - 15:30! ! ! Questions and comments

June, 30th ! ! 15:30 - 15:45! ! ! Numerical implementation of turbulent flow models

Alfonso BUENO - bueno@caminos.upm.es

Abstract:

It will be discussed the numerical implementation and possible applications of turbulent flow models in applied shape optimization for CFD. Specifically, the presentation will be focused in the Spalart-Allmarat model and its potential use in optimal aeronautics design.

13

! ! !! ! ! 15:45 - 16:00! ! ! Questions and comments

June, 30th ! ! 16:00 - 16:30! ! ! COFFEE BREAK

BCAM Headquarters

14

June, 30th ! ! 16:30 - 16:45! ! ! Some problems related with homogeneous kinetic equations

Miguel ESCOBEDO - miguel.escobedo@ehu.es

Abstract:

I will briefly expose some of the questions that I have been considering, related with spatially homogeneous kinetic equations and qualitative properties of their solutions. I will also try to describe possible future interactions with other topics in the project.! ! !! ! ! 16:45 - 17:00! ! ! Questions and comments

15

June, 30th ! ! 17:00 - 17:15! ! ! Hardy-Poincare inequalities with singularities on the boundary and applications in the control of the wave equation with boundary singular potential

Cristian CAZACU - cazacu@bcamath.org

Abstract:

In this talk we are mainly interested in optimal Hardy-Poincare inequalities with singularities on the boundary. We are able to prove that the optimality of these inequalities depends on the geometry of the domain considered. More precisely, we will see that some regularity of the boundary and the sign of the mean curvature at the singular point, play a crucial role to establish the optimality. As an application, we prove the null-controllability of the wave equation with boundary singular potential. Due to the well-known method of multipliers, the control results for the wave equation can be extended in the range of lambda's for which the Hardy inequality holds.

! ! ! 17:15 - 17:30! ! ! Questions and comments

16

June, 30th ! ! 20:10! ! ! DINNER: CHINESSE RESTAURANT “ERCILLA”

C/ Ercilla, 15 BILBAO. HOW TO REACH FROM HOTEL ABBA

17

July, 1st ! ! 11:00 - 11:15! ! ! Periodic controllability of evolution equations

Xu ZHANG - xuzhang@amss.ac.cn

Abstract:

This work is addressed to a study of the periodic controllability of linear evolution equations in Hilbert space. Equivalent conditions for this property are presented, which lead to some new observability estimates for uncontrolled systems. Moreover, periodic controllability for the wave equations will be analyzed in detail, and a list of open problems will be discussed.

! ! !! ! ! 11:15 - 11:30! ! ! Questions and comments

18

July, 1st ! ! 11:30 - 12:00! ! ! COFFEE BREAK

BCAM Headquarters

19

July, 1st ! ! 12:00 - 12:15! ! ! MTM Activity Report

Liviu IGNAT - liviu.ignat@gmail.com

Abstract:

1. Schrödinger equations on trees 2. Splitting methods for Nonlinear Schrödinger equation

! ! !! ! ! 12:15 - 12:30! ! ! Questions and comments

20

July, 1st ! ! 12:30 - 12:45! ! ! Nonconforming and Discontinuous Galerkin Finite Element Methods

Blanca AYUSO - blanca.ayuso@uam.es

Abstract:

Research interests: Main research field: numerical analysis, discretization of partial differential equations, finite elements techniques.Main keywords: Basic Properties of Finite element methods; conforming nonconforming and mixed, Discontinuous Galerkin Methods, Nonconforming domain decomposition methods, Multigrid methods, Kinetic equations, Linear Elasticity, Advection diffusion-reaction and hyperbolic problems.(in the past also: Stabilization techniques for finite element discretizations, Postprocessing techniques, Navier-Stokes equations)

! ! !! ! ! 12:45 - 13:00! ! ! Questions and comments

21

June, 1st ! ! 13:00 - 15:00! ! ! LUNCH

BCAM Headquarters

22

July, 1st ! ! 15:00 - 15:15! ! ! Desarrollo de un Método de Elementos Finitos para Realizar Simulaciones Multifísicas de Gran Precisión

Iñaki GARAY - garay@bcamath.org

Abstract:

Para determinar las características de una reserva petrolífera (o para la detección de cáncer), es usual invertir mediciones geofísicas (o médicas) que se obtienen utilizando distintos tipos de sensores (transmisores y receptores). Dichos sensores son normalmente colocados en distintas posiciones, adquiriendose así múltiples mediciones que, en general, se basan en distintos fenómenos físicos tales como acústica, electromagnetismo y fenómenos nucleares. Cuando simulamos mediciones de cualquier tipo, las interacciones entre distintas físicas suelen ser ignoradas, ya que dichas interacciones son débiles y difíciles de interpretar físicamente. Sin embargo, las interacciones existentes en problemas inversos entre físicas de distinta naturaleza son mucho más intensas que en problemas directos. De hecho, es posible observar una gran correlación entre las mediciones correspondientes a distintas físicas, como queda claramente reflejado, por ejemplo, en las mediciones sóicas y electromagnéticas que se obtienen en pozos petrolíferos o para la detección de cáncer.

23

En nuestra opinión, estas correlaciones deben ser estudiadas en detalle, ya que facilitan la correcta caracterización de una reserva petrolífera o de un posible tumor (forma, localización, extensión, y características físicas). Además, estas interacciones entre distintas físicas pueden ser utilizadas para reducir la no unicidad del problema inverso.En esta presentación describiremos una infraestructura de software que estamos desarrollando para la simulación e inversión de fenómenos multifísicos, lo que permitirá la inversión conjunta de procesos multifísicos para la correcta caracterización de distintos materiales vía mediciones no invasivas. Esta infraestructura está fundamentada en diversos conceptos matemáticos que explicaremos en detalle, tales como el diagrama de Rham, elementos finitos de orden superior, y el problema adjunto, y cuenta con sofisticados algoritmos para su ejecución en ordenadores en paralelo, y la correcta generación de mallados óptimos, lo que permite obtener soluciones de gran precisión. También mostraremos resultados preliminares de aplicaciones de este método a la correcta caracterización de pozos petrolíferos.

! ! !! ! ! 15:15 - 15:30! ! ! Questions and comments

24

July, 1st ! ! 15:30 - 15:45! ! ! Traveling waves for models of phase transitions of solids driven by configurational forces

Peicheng ZHU - zhu@bcamath.org

Abstract:

My talk is concerned with the existence of traveling wave solutions, including standing waves, to some models based on configurational forces, describing respectively the diffusionless phase transformations of solids, e.g., Steel, and phase transitions due to interface motion by interface diffusion, e.g., Sintering. Also we compare our results with the corresponding ones for the Allen-Cahn and the Cahn-Hilliard equations coupled with linear elasticity, which are models for diffusion-dominated phase transformations in elastic solids.

! ! !! ! ! 15:45 - 16:00! ! ! Questions and comments

25

July, 1st ! ! 16:00 - 16:15! ! ! Traveling waves for models of phase transitions of solids driven by configurational forces

Vincent LESCARRET - lescarret@bcamath.org

Abstract:

We consider the propagation of waves in the multi-structure studied by H. Koch and E. Zuazua, on the numerical point view. The multi-structure is modelized by two disjoint media separated by a fixed weighted interface and the motion of the wave is described by a wave equation in each of the three components of the domain with Dirichlet boundary conditions. We consider the finite-difference discretisation of this problem and show that for a straight interface the numerical waves exist in some spaces with asymmetric regularity. Numerical computations are also provided.

! ! !! ! ! 16:15 - 16:30! ! ! Questions and comments

26

June, 1st ! ! 16:30 - 17:00! ! ! COFFEE BREAK AND CONCLUSIONS

BCAM Headquarters

27