Workshop on APPLIED MATHEMATICSestimates of order hk and hk+1/2 using a suitably stabilized...

W o r k s h o p o n A P P L I E D M A T H E M A T I C SHOTEL INDAUTXU - B I LBAO

July, 2nd - 4 th, 2009

Organizers & Scientists

http://www.bcamath.org/bcam-cim-2009 2

ORGANIZERSORGANIZERS

José Miguel Urbano, CMUC jmurb@mat.uc.pt

Enrique Zuazua, Ikerbasque - BCAM zuazua@bcamath.org

SCIENTISTSSCIENTISTSSCIENTISTSSCIENTISTS

Miguel Escobedo, EHU-UPV miguel.escobedo@ehu.es Alexis F. Vasseur, U. Texas Austin vasseur@math.utexas.edu

Irene Fonseca, Carnegie Mellon fonseca@andrew.cmu.edu Juha Videman,IST jvideman@math.ist.utl.pt

Manuel Luna, U. Sevilla mllainez@us.es Peicheng Zhu, Ikerbasque - BCAM zhu@bcamath.org

Ander Murua, EHU-UPV ander.murua@ehu.es Vincent Lescarret, BCAM lescarret@bcamath.org

Francisco Palacios, UPM fpalacios@gmail.com Aurora Marica, BCAM marica@bcamath.org

José F. Rodrigues, UL & CMAF rodrigue@ptmat.fc.ul.pt Carlos Mora-Corral, BCAM mora@bcamath.org

Julio Rossi, FCyN & UBA jrossi@dm.uba.ar Gonçalo Pena, CMUC gpena@mat.uc.pt

Sponsors / Dates / Venue

SPONSORS• Basque Government - Ministry of Education, Universities and Research• CIM - Centro Internacional de Matematica• BCAM - Basque Center for Applied Mathematics• MTM2008-03541 Project• SEMA - Sociedad Española de Matemática Aplicada

2nd - 4th, July, 2009

HOTEL INDAUTXU, C/ Bombero Etxaniz, s/n BILBAO

Program schedule

July, 2nd ! !13:00 - 13:30R e g i s t r a t i o n

• HOTEL INDAUTXU

July, 2nd ! !13:30 - 15:00L U N C H

• HOTEL INDAUTXU

July, 2nd ! !15:00 - 15:15! ! !Presentation of CIM

José F. RODRIGUES - rodrigue@ptmat.fc.ul.pt

Presentation of CIM - Centro Internacional de Matematica (Portugal)

July, 2nd ! !15:15 - 15:30! ! !Presentation of BCAM

Enrique ZUAZUA - zuazua@bcamath.org

Presentation of BCAM - Basque Center for Applied Mathematics (Basque Country)

July, 2nd ! !15:30 - 16:15! ! !Variational Reconstruction of Color Images

Irene FONSECA - fonseca@andrew.cmu.edu

Abstract:In this talk we will use recently developed variational techniques for the restoration of damaged frescoes to recover full colorization of an image for which the underlying gray level function is known everywhere although only few sparse regions of color are available. This is joint work with Giovanni Leoni, Francesco Maggi and Massimiliano Morini.

July, 2nd ! !16:15 - 17:00Non zero flux solutions for some homogeneous kinetic equations

Miguel ESCOBEDO - miguel.escobedo@ehu.es

Abstract:Boltzmann type kinetic equations present some formal conservation laws, reflecting the underlying physical hypothesis. We will see that some of these equations have classical solutions which do not satisfy these conservation laws and are physically admissible.

July, 2nd ! !17:30 - 18:00Fluid-structure interaction: a spectral approach

Gonçalo PENA - gpena@mat.uc.pt

Abstract:Accuracy is critical if we are to trust simulation predictions. In settings such as fluid-structure interaction it is all the more important to obtain reliable results to understand, for example, the impact of pathologies on blood flows in the cardiovascular system. In this talk, we propose a computational strategy for simulating fluid structure interaction using high order methods in space and time.First, we present the mathematical and computational core framework, Life, underlying our multi-physics solvers. Life is a versatile library allowing for 1D, 2D and 3D partial differential solves using $h/p$ type Galerkin methods. Then, we briefly describe the handling of high order geometry and the structure solver. Next we outline the high-order space-time approximation of the incompressible Navier-Stokes equations and comment on the algebraic system and the preconditioning strategy. Finally, we present the high-order Arbitrary Lagrangian Eulerian (ALE) framework in which we solve the fluid-structure interaction problem as well as some results.

July, 2nd ! !18:00 - 18:30Concentrated 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 increase polynomially with an order depending on the regularity of the

Fourier transform of the initial data.

July, 3rd ! !9:30 - 10:15Recent results in fluid mechanics

Alexis F. VASSEUR - vasseur@math.utexas.edu

Abstract:We will present, in this talk, new applications of De Giorgi's methods and blow-up techniques to fluid mechanics problems. Those techniques have been successfully applied to show full regularity of the solutions to the surface quasi-geostrophic equation in the critical case.We will present, also, a new nonlinear family of spaces allowing to control higher derivatives of solutions to the 3D Navier-Stokes equation. Finally, we will present a regularity result for a reaction-diffusion system which has almost the same supercriticality than the 3D Navier-Stokes equation.

July, 3rd ! !10:15 - 11:00Splitting methods for the time integration of wave equations

Ander MURUA - ander.murua@ehu.es

Abstract:Splitting methods for the time integration of differential equations can be successfully applied in a great variety of initial valued problems. They are easy to implement, often preserve qualitative properties of the original problem, and can be very efficient. Here, we consider the application of splitting methods to linear systems of ordinary differential equations of the form

(H a real symmetric matrix) that arise in the spatial semidiscretization of the wave equation. Similar techniques can be applied for the time integration of Schrödinger equations.

dt2 u = !Hu

July, 3rd ! !11:30 - 12:15Streamline diffusion methods for Shallow-Water Equations

Juha VIDEMAN - jvideman@math.ist.utl.pt

Abstract:We consider streamline diffusion, also known as SUPG (Streamline Upwind Petrov–Galerkin), methods applied to the time-dependent shallow-water equations. Streamline diffusion (SD) methods are finite element methods that combine good stability properties with high accuracy and are particularly suitable for hyperbolic and advection-diffusion equations. The SUPG method, introduced by Thomas Hughes and Alexander Brooks in 1979, was applied and analysed intensively throughout the 80’s by Hughes and, in parallel, by Claes Johnson, and their co-workers. Johnson adopted the name streamline diffusion method, extended it to the time–dependent problems and related the method, regarding the time discretization, to the discontinuous Galerkin method.

Written in conservation form (mass/momentum flux), the shallow-water equations constitute a non-linear hyperbolic system, similar to the compressible Navier-Stokes equations, and their numerical approximation, either in conservative or non-conservative form, has been obtained by various finite difference and finite element methods, most recently by local discontinuous Galerkin methods, cf. Rigorous error analyses have, however, been scarce and even more so for the fully discretized problem written in terms of the non-conservative variables – the depth-integrated horizontal velocities and the height of the free surface.

In this talk, I will present some of our recent results on the application of SD methods, with time–space elements, to two–dimensional shallow-water equations. We will prove error estimates of order hk and hk+1/2 using a suitably stabilized variational formulation. Our finite element approximation is continuous in space but possibly discontinuous in time and we use kth–order polynomials for the surface height and polynomials of order k or k + 1 for the velocities.

This is a joint work with Clint Dawson from the Center for Subsurface Modeling at the Institute for Computational Engineering and Sciences at the University of Texas at Austin (USA).

July, 3rd ! !12:15 - 13:00Toward real-time aerodynamic simulation and immersive visualization

Francisco PALACIOS - fpalacios@gmail.com

Abstract:Numerical Simulation has been acquiring increasing relevance within the aeronautical community, both in the industrial sector as well as in research centers. Unfortunately the current technology doesn’t provide us enough computational power to develop real time simulation or a real immersive visualization which make easier the interaction between the engineer and the machine.In this presentation I will review some results obtained within the framework of the FuSim-E Programme. This programme is set to transform simulation and design processes for the aerodynamics through the development of innovative computer based simulation systems upon solid mathematical bases. As a result of the more efficient and powerful simulation and design processes we expect that more effort can be focused on the development of environmentally friendly products. In this presentation I’ll show some mathematical-computational techniques which can make a change in the today simulation and design conceps: adjoint techniques, surface parameterization, hardware acceleration, domain decomposition, massive parallelization with real time monitoring, grid technology, virtual reality, among other experimental techniques developed within FuSim-E programme.

July, 3rd ! !13:00 - 16:00W O R K S H O P L U N C H

• HOTEL INDAUTXU

July, 3rd ! !16:00 - 16:30Variational models for cavitation and fracture in nonlinear elasticity

Carlos MORA-CORRAL - mora@bcamath.org

Abstract:First we review the standard variational models for cavitation and for fracture in nonlinear elasticity, in the static case. Then we formulate and analyze our existence theory for deformations that may undergo both cavitation and fracture. That theory has some connections with abstract results on weak continuity of the determinant and on regularity of the inverse of invertible BV maps. This is a joint work with Duvan Heano.

July, 3rd ! !16:30 - 17:15Traveling waves to models for phase transitions in solids driven by configurational forces

Peicheng ZHU - zhu@bcamath.org

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

July, 3rd ! !17:45 - 18:15Numerical analysis of waves propagating in some multi-structure

Vincent LESCARRET - lescarret@bcamath.org

Abstract:We consider the propagation of waves in the multi-structure studied by Koch-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 discretization 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.

July, 3rd ! !21:00 -W O R K S H O P D I N N E R

• Sidrería Asador ARRIAGA C/Santa María, 13 E-48005 BILBAO

July, 4th ! !10:00 - 10:45Tug-of-War games and PDE’s

Julio ROSSI - jrossi@dm.uba.ar

Abstract:In this talk we characterize p-harmonic functions in terms of an asymptotic mean value property. A p-harmonic function u is a viscosity solution to ∆pu=div(|∇u|p-2∇u)=0

holds as ε→0 for x∈Ω holds in a weak sense, which call viscosity sense. Here the coe-

fficients α, β are determined by α+β=1, and α/β=(p-2)/(N+2).This asymptotic mean value formula is inspired by game theory when one considers values of Tug-of-War games.

Joint work with J. J. Manfredi and M. Parviainen.

u(x) = !2

max uB!(x) + min u

B!(x) + "|B!(x)|

!B!(x) udy + o(!2)

July, 4th ! !10:45 - 10:45On the slip condition for a viscous fluid near a rough wall with very small asperities

Manuel LUNA - mllainez@us.es

Abstract:We study the asymptotic behavior of a viscous fluid near a periodic oscillating wall Γ ε of period ε and amplitude δε , when ε and δε tend to zero. We assume the fluid to satisfy the slip condition given by Navierʼs law

uε · n = 0, (Duε · n)tan + κuε = 0 on Γε , where uε is the velocity, n is a normal vector to Γε , κ ≥ 0 is the friction coefficient and the subscript tan denotes the tangential pro jection. That is, the fluid can not go across the wall, which is not permeable to the fluid particles, but can slide on it. When δε is of order ε, it is known that in the limit this boundary condition provides the adherence (or no-slip) condition u = 0 on Γ, where Γ is the limit plane wall of Γε . This gives a mathematical justification of why adherence conditions are usually imposed for viscous fluids, they may be due to the microscopic asperities. In this work we consider the case where δε is much smaller than ε. We show that if δε /ε3/2 tends to infinity, then we still obtain an adherence condition in the limit. If δε /ε3/2 tends to zero, the rugosity of the wall is worthless and the fluid behaves as if Γε was a plane wall. Finally, if the limit of δε /ε3/2 is in (0, +∞) (critical case) then, although the rugosity is not big enough to give an adherence condition, we obtain a new Navierʼs law with a larger friction coefficient. In general, this new friction coefficient is not a scalar but a matrix.

July, 4th ! !12:00 - 12:45Quasi-variational solutions to first order quasilinear equations with gradient constraint

José Francisco RODRIGUES - rodrigue@ptmat.fc.ul.pt

Abstract:We prove the existence of variational solutions for an evolution quasi-variational inequality with a first order quasilinear operator and a variable convex set which is characterized by a cons-traint on the absolute value of the gradient that depends on the solution itself. The only requi-red assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable "a priori" estimates and allow us to obtain also the existence of stationary solutions, by studying the asymptotic behaviour in time. This is a joint work with Lisa Santos.

July, 4th ! !13:30 - L U N C H

• HOTEL INDAUTXU

Workshop on APPLIED MATHEMATICSestimates of order hk and hk+1/2 using a suitably stabilized...

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