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1. Happy birthday to Ricardo and welcome to Gargnano
In occasion of Ricardo H. Nochetto’s 60th Birthday, his friends, colleagues and students hope he, and all
the other, will enjoy this small-scale workshop focusing on recent advances in the Numerical Approximation
of Partial Differential Equations in relation to Nochetto’s work.
Adaptive finite element methods are a fundamental numerical tool in science and engineering. They are
known to outperform classical finite element in practice and deliver optimal convergence rates when the latter
cannot. The aim of this workshop is to stimulate a fruitful discussion regarding adaptivity, error control and
convergence analysis in the context of numerical approximation of PDEs.
The workshop will take place on March 20-22, 2013, at the prestigious Palazzo Feltrinelli, 18 Via XXIV
Maggio, Gargnano del Garda, Brescia (Italy). More information can be found on the website http://www.
napde.org/
This meeting has been set up by the members of the Organizing Committee (OC): Andrea Bonito, Omar
Lakkis, Pedro Morin, Andreas Veeser, Marco Verani, Claudio Verdi, Chen-Song Zhang.
The OC acknowledges Giuseppe Savaré’s very important contribution and suggestions. Special thanks go
to Marie Bonito for the design of the poster.
The Scientific Committee (SC) is formed by: Claudio Canuto, Zhi-Ming Chen, Ronald DeVore, Vivette
Girault, Charalambos G. Makridakis, Giuseppe Savaré, Kunibert G. Siebert, Andreas Veeser, Pedro Morin,
Claudio Verdi.
The University of Milano and MOX logistic and financial support was essential and is throughly acknowledged.
Hopefully this booklet contains all the important information for a smooth running of events, but it is
very likely that something was missed out. In that case, if you need assistance, please refer directly to that
member of the OC that is closest to you.
Contents
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2.
3.
4.
5.
6.
7.
Happy birthday to Ricardo and welcome to Gargnano
Travel information
Maps
Workshop information
Participants
Schedule
Talk data
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2. Travel information
The following information on how to reach Gargnano has been copy-pasted from
http://www.gargnanosulgarda.it/tourist-service-lakegarda/how-to-reach-gargnano.html
where you will find useful links to various services. For the return trip, invert the route.
2.1. By train or bus.
From Milano (Milan). Get to Brescia Railway Station. Outside the Station Building, 200 meters in front
around the corner on the left, there’s the Bus Stand to the Lake (not the round one on the right), whose
last stop is Gargnano is case of difficulty, ask for “ autobus per il lago”.
From Venezia (Venice). Go to Brescia and follow previous instructions or stop at Desenzano Railway station
and get a bus to Gargnano by another line. Another option is to join any location on the lake and get a
boat to Gargnano
From Rovereto. Get to Riva del Garda with this bus line. Once in Riva, get a bus to Desenzano, which stops
in Gargnano, with this line. You can also take a boat transportation from Riva del Garda to Gargnano.
2.2. By car.
From Milano. Take Highway A4 Milano - Venezia, exit Brescia Est and head to Salò. Then take Riva del
Garda direction and after average 13 km you will find Bogliaco, Villa and last, Gargnano Center. Take the
only road entering the center.
From Venice. Take Highway A4 Venezia-Milano, exit at Desenzano del Garda and go to Salò.Then take Riva
del Garda direction and after average 13 km you will find Bogliaco, Villa and last, Gargnano Center.
From Rovereto. Take Riva del Garda direction and then head to Brescia. after average 30 km you will find
Gargnano. U turn to enter in the center.
2.3. From Malpensa (Milan) airport. Bus/Train: just out of the terminal take the Malpensa Shuttle to
Milano Centrale RLW Station, catch the first train to Brescia, once out Brescia RLW Station, 200 meters
front left, take the bus to the lake.The last stop is Gargnano,
Car: Take the highway, direction Venezia, exit at Brescia Est, heading to Salò. Then take the direction
Riva del Garda. After around 15 km, you find Bogliaco, then Villa and then Gargnano Centro. Take the
only road entering the center.
2.4. From Linate (Milan) airport. Bus/Train: just out of the terminal take any public mean to join
Milano Centrale RLW Station, catch the first train to Brescia, once out Brescia RLW Station, 200 meters
front left, take the bus to the lake.The last stop is Gargnano.
Car: Take the highway, direction Venezia, exit at Brescia Est, heading to Salò. From Salò, take the
direction Riva del Garda. After around 15 km, you find Bogliaco, then Villa and then Gargnano Centro.
Take the only road entering the center.
2.5. From Bergamo airport. Bus/Train: from the airport, with the Shuttle Bus,join Bergamo RLW
Station and from there, take the train to Brescia, once out Brescia RLW Station, 200 meters front left, take
the bus to the lake. The last stop is Gargnano.
Car: Take the highway, direction Venezia, exit at Brescia Est, heading to Salò. From Salò, take the
direction Riva del Garda. After around 15 km, you find Bogliaco, then Villa and then Gargnano Centro.
Take the only road entering the center.
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2.6. From Verona “Catullo” airport. Boat: from any locality of the lake, you can take the service boat
of Navigarda and join Gargnano.
Bus/Train: Catch any public mean to join Verona RLW Station and from there, take the train to Desenzano, and then the bus to Gargnano or join Brescia RLW Station.Once out, 200 meters front left, take the
bus to the lake. The last stop is Gargnano, 100 meters from our hotel.
Car: Take the highway, direction Milano, exit at Desenzano, heading to Salò. From Salò, take the direction
Riva del Garda. After around 15 km, you find Bogliaco, then Villa and then Gargnano Centro. Take the
only road entering the center.
2.7. From Montichiari “D’Annunzio” airport. Boat: join any locality of the lake, from there you can
take the service boat of Navigarda and join Gargnano.
Bus: Catch any public mean to join Brescia RLW Station. There, take the bus to the lake. Car: Head to
Salò. Then take the direction Riva del Garda. After around 15 km, you find Bogliaco, then Villa and then
Gargnano Centro. Take the only road entering the center.
2.8. From Venice. Bus/Train: From Venezia S.Lucia RLW Station, take the train to Desenzano, and then
the bus to Gargnano or join Brescia RLW Station. Once out, 200 meters front left, take the bus to the lake.
The last stop is Gargnano.
Car: Take the highway, direction Milano, exit at Desenzano, heading to Salò. From Salò, take the direction
Riva del Garda. After around 15 km, you find Bogliaco, then Villa and then Gargnano Centro. Take the
only road entering the center.
2.9. From Rovereto railway station. Bus: take the bus Rovereto-Riva del Garda from here, catch the
bus to Desenzano. Gargnano is roughly half way.
Car: take the direction of Riva del Garda, then head to Brescia and after average 30 km you find Gargnano.
To enter in the center, you must make a U turn.
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3. Maps
Gargnano is located on the west shore of
the Lake Garda, in the Italian Province of
Brescia, in the middle of the Regional Park
“Alto Garda”. It has a very interesting historic background, including the history of
Palazzo Feltrinelli where the workshop is
taking place. Lake Garda being a prime
tourist destination there are plenty of extraacademic activities for you to partake in
from windsurfing to hiking in the park.
Gargnano can be reached by road and by
boat. The closest major railway stations are
Desenzano (south-east) and Brescia (southwest). It is also possible to reach it by boat.
Check the travel information for more.
Gargnano is quite small and cosy. From
the bus station to Palazzo Feltrinelli is 5 a
minutes walk outlined in this map.
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4. Workshop information
4.1. Registration. Registration will open on Tuesday 19 March at 15:00 and will be at Palazzo Feltrinelli.
The address of Palazzo Feltrinelli is Via XXIV Maggio 18, Gargnano.
4.2. Payment. Payments for meals and accommodations in Palazzo Feltrinelli/Casa Bertolini: please remember the non-invited (’not yellow’) that the payments for the meals and the accommodation in Palazzo
Feltrinelli and Casa Bertolini has to be done in cash. The rooms of Palazzo Feltrinelli and Casa Bertolini
have to be left on the departure day by 9:00 (otherwise another day will be charged). People with accommodation at Casa Bertolini (50 m from the Palazzo) have to register at Palazzo Feltrinelli in order to get
the key of their room.
4.3. Accommodation. You should by now have all the details of your accommodation which, barring
personal arrangements, is located in one of Palazzo Feltrinelli, Casa Bertolini or Hotel Meandro.
4.4. Lunches and dinners. These are served in Palazzo Feltrinelli. The times are in the schedule. Please
be on time to avoid missing out. According to the house rules, you must be registered for each lunch or
dinner you are planning to have. Should the information you received be different, please contact a member
of the OC immediately.
4.5. Conference dinner. The conference dinner, and the friendly celebration of Ricardo’s achievement,
will take place on Thursday evening.
4.6. Own device connection. There is a wireless connection in Palazzo Feltrinelli. To obtain access you
will need to present a valid ID (e.g., passport) a copy of which has to be made by the staff (Italian law). In
Casa Bertolini, there is unfortunately, no direct wireless access. In the Hotel Meandro, you should ask the
hotel staff.
4.7. Audiovisual aids. All lectures and talks will take place in the “Aula magna”. We are planning to
arrange a single computer-projector set-up, so speakers should have a PDF-ready talk (if you have more
complicated presentations, e.g., movies, non-PDF format, etc. please let the organizers know as early as you
can).
4.8. Other questions? Ask a member of the OC.
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5. Participants
Georgios Akrivis: akrivis@cs.uoi.gr
University of Ioannina
Department of Computer Science
Ioannina, Greece
Harbir Antil: hantil@gmu.edu
George Mason University
Department of Mathematical Sciences
Fairfax, Virginia USA
Doug Arnold: arnold@umn.edu
University of Minnesota
School of Mathematics
Minneapolis, Minnesota USA
Ayuso de Dios Blanca: blanca2877@gmail.com
Centre de Recerca Matemàtica
Bellaterra
Barcelona, Catalunya, Spain
Eberhard Bänsch: baensch@math.fau.de
University of Erlangen
Department Mathematik
Erlangen, Germany
Soeren Bartels: bartels@mathematik.uni-freiburg.de
University of Freiburg
Department Angewandte Mathematik
Freiburg, Germany
Stefano Berrone: sberrone@calvino.polito.it
Politecnico di Torino
Dipartimento di Scienze Matematiche
Torino, Italy
Daniele Boffi: daniele.boffi@unipv.it
University of Pavia
Dipartimento di Matematica
Pavia, Italy
Andrea Bonito: bonito@math.tamu.edu
Texas A & M University
Department of Mathematics
College Station, Texas USA
Andrea Bressan: andrea.bressan@unipv.it
Università di Pavia
Dipartimento di Matematica
Pavia, Italy
Luis Caffarelli: caffarel@math.utexas.edu
University of Texas Austin
Department of Mathematics
Austin, Texas USA
Claudio Canuto: claudio.canuto@polito.it
Politecnico di Torino
Dipartimento di Scienze Matematiche
Torino, Italy
Lara Antonella Charawi: lara.charawi@unimi.it
Università degli Studi di Milano
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Dipartimento di Matematica
Milano, Italy
Zhiming Chen: zmchen@lsec.cc.ac.cn
Chinese Academy of Sciences
Academy of Mathematics and Systems Science
Beijing, China
Long Chen: chenlong@math.uci.edu
University of California at Irvine
Department of Mathematics
Irvine, California USA
Bernardo Cockburn: cockburn@math.umn.edu
University of Minnesota
School of Mathematics
Minneapolis, Minnesota USA
Piero Colli Franzone: colli@imati.cnr.it
Università di Pavia
Dipartimento di Matematica
Pavia, Italia
Ronald DeVore: ronald.a.devore@gmail.com
Texas A & M University
Department of Mathematics
College Station, Texas USA
Georg Dolzmann: Georg.Dolzmann@mathematik.uni-regensburg.de
Universität Regensburg
Fakultät für Mathematik
Regensburg, Germany
Willy Dörfler: willy.doerfler@kit.edu
Karlsruhe Institute of Technology
Department of Mathematics
Karlsruhe, Germany
Fernando Gaspoz: fernando.gaspoz@ians.uni-stuttgart.de
Universität Stuttgart
Instituts für Angewandte Analysis und Numerische Simulation
Stuttgart, Germany
Lucia Gastaldi: lucia.gastaldi@ing.unibs.it
University of Brescia
Dipartimento di Matematica
Brescia, Italy
Fotini Karakatsani: fotkara@gmail.com
University of Strathclyde
Mathematics and Statistics
Glasgow, Scotland UK
Theodoros Katsaounis: thodoros65@gmail.com
Univerisity of Crete
Department of Applied Mathematics
Heraklion, Greece
Christian Kreuzer: christian.kreuzer@rub.de
Ruhr Universität Bochum
Fakultät für Mathematik
Bochum, Germany
Irene Kyza: kyza@iacm.forth.gr
University of Dundee
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Division of Mathematics
Dundee, Scotland UK
Omar Lakkis: o.lakkis@sussex.ac.uk
University of Sussex
Department of Mathematics
Brighton, England UK
Charalambos Makridakis: makr@tem.uoc.gr
University of Crete
Department of Applied Mathematics
Heraklion, Greece
Giovanni Migliorati: giovanni.migliorati@gmail.com
École Polythechnique Fédérale de Lausanne
MATHICSE
Lausanne, Switzerland
Pedro Morin: morinpedro@gmail.com
Universidad Nacional del Litoral
IMAL, CONICET
Santa Fe, Argentina
Ricardo Nochetto: rhn@math.umd.edu
University of Maryland
Department of Mathematics & IPST
College Park, Maryland USA
Maurizio Paolini: paolini@dmf.unicatt.it
Universita’ Cattolica Brescia
Dipartimento di Matematica e Fisica
Brescia, Italy
Sebastian Pauletti: spauletti@gmail.com
Consiglio Nazionale delle Ricerche (CNR)
IMATI ’Enrico Magenes’
Pavia, Italy
Ilaria Perugia: ilaria.perugia@unipv.it
Università di Pavia
Dipartimento di Matematica
Pavia, Italy
Paola Pietra: pietra@imati.cnr.it
Consiglio Nazionale delle Ricerche (CNR)
IMATI ’Enrico Magenes’
Pavia, Italy
Jae-Hong Pyo: jhpyo@kangwon.ac.kr
Kangwon National University
Department of Mathematics
Chuncheon, Korea
Rodolfo Rodriguez: rodolfo@ing-mat.udec.cl
Universidad de Concepcion
Departamento de Ingenieria Matematica
Concepcion, Chile
Sandro Salsa: sandro.salsa@polimi.it
Politecnico di Milano
Dipartimento di Matematica ””Francesco Brioschi””
Milano, Italy
Giuseppe Savaré: giuseppe.savare@unipv.it
University of Pavia
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Department of Mathematics
Pavia, Italy
Alfred Schmidt: schmidt@math.uni-bremen.de
Universität Bremen
Zentrum für Technomathematik
Bremen, Germany
Natasha Sharma: natasha.sharma@iwr.uni-heidelberg.de
University of Heidelberg
IWR
Heidelberg, Germany
Kunibert G. Siebert: kg.siebert@ians.uni-stuttgart.de
Universität Stuttgart
Instituts für Angewandte Analysis und Numerische Simulation
Stuttgart, Germany
Francesca Tantardini: francesca.tantardini1@unimi.it
Università di Milano
Dipartimento di Matematica
Milano, Italy
Andreas Veeser: andreas.veeser@unimi.it
Università di Milano
Dipartimento di Matematica
Milano, Italy
Marco Verani: marco.verani@polimi.it
Politecnico di Milano
Dipartimento di Matematica
Milano, Italy
Claudio Verdi: claudio.verdi@unimi.it
Università di Milano
Dipartimento di Matematica
Milano, Italy
Shawn Walker: walker@math.lsu.edu
Louisiana State University
Department of Mathematics
Baton Rouge, Louisiana USA
Chensong Zhang: zhangchensong@gmail.com
Chinese Academy of Sciences
Academy of Mathematics and System Sciences
Beijing, China
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6. Schedule
15:00 to 19:30
19:30 to 21:30
08:45
09:00
10:00
10:35
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11:35
12:10
12:45
15:30
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21:30
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20:30
Tuesday, 19 March 2013
registration
dinner
Wednesday, 20 March 2013
welcome
keynote talk by Ronald DeVore
(T12)
invited talk by Georgios Akrivis
(T1)
coffee
invited talk by Eberhard Bänsch
(T4)
invited talk by Zhiming Chen
(T9)
invited talk by Irene Kyza
(T17)
lunch
invited talk by Andrea Bonito
(T6)
invited talk by Marco Verani
(T24)
coffee
keynote talk by Bernardo Cockburn
(T11)
invited talk by Christian Kreuzer
(T16)
social
dinner
Thursday, 21 March 2013
keynote talk by Luis Caffarelli
(T7)
invited talk by Rodolfo Rodriguez
(T20)
coffee
invited talk by Lucia Gastaldi
(T15)
invited talk by Harbir Antil
(T2)
invited talk by Sören Bartels
(T5)
lunch
invited talk by Long Chen
(T10)
invited talk by Jae-Hong Pyo
(T19)
coffee
keynote talk by Douglas Arnold
(T3)
invited talk by Shawn Walker
(T25)
social
conference dinner followed by party
Friday, 22 March 2013
keynote talk by Sandro Salsa
(T21)
invited talk by Alfred Schmidt
(T22)
coffee
invited talk by Chensong Zhang
(T26)
invited talk by Kunibert G. Siebert
(T23)
lunch
invited talk by Georg Dolzmann
(T13)
invited talk by Omar Lakkis
(T18)
coffee
invited talk by Willy Dörfler
(T14)
invited talk by Claudio Canuto
(T8)
dinner
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7. Talk data
T1. Georgios Akrivis. Implicit-explicit multistep methods for a class of nonlinear parabolic equations
Abstract: We consider the discretization of an initial value problem for a nonlinear parabolic equation,
in an abstract Hilbert space setting, by combinations of implicit and explicit multistep schemes. We will
discuss consistency and stability of the schemes, under certain conditions, and will derive optimal order
error estimates. The stability assumptions can be relaxed in the case of first and second order schemes. The
discretization of nonlinear convection-diffusion equations by implicit-explicit multistep schemes will also be
briefly discussed.
T2. Harbir Antil. A Stokes free boundary problem with surface tension effects
Abstract: We consider a Stokes free boundary problem with surface tension effects in variational form. This
model is an extension of the coupled system proposed by P. Saavedra and L. R. Scott, where they consider
a Laplace equation in the bulk with Young-Laplace equation on the free boundary to account for surface
tension. The two main difficulties for the Stokes free boundary problem are: the vector curvature on the
interface, which causes problem to write a variational form of the free boundary problem and the existence of
1+1/p′
solution to Stokes equations with Navier-slip boundary conditions for Wp
domains (minimal regularity).
We will demonstrate the existence of solution to Stokes equations with Navier-slip boundary conditions using
1+1/p′
a perturbation argument for the bended half space followed by standard localization technique. The Wp
regularity of the interface allows us to write the variational form for the entire free boundary problem, we
conclude with the well-posedness of this system using a fixed point iteration.
T3. Doug Arnold. The periodic table of finite elements
Abstract: Finite element methodology, reinforced by deep mathematical analysis, provides one of the most
important and powerful toolsets for numerical simulation. Over the past forty years a bewildering variety
of different finite element spaces have been invented to meet the demands of many different problems. The
relationship between these finite elements has often not been clear, and the techniques developed to analyze
them can seem like a collection of ad hoc tricks. The finite element exterior calculus, developed over the
last decade, has elucidated the requirements for stable finite element methods for a large class of problems,
clarifying and unifying this zoo of methods, and enabling the development of new finite elements suited to
previously intractable problems. In this talk, we will discuss the big picture that emerges, providing a sort
of periodic table of finite element methods.
T4. Eberhard Bänsch. A posteriori error estimates for approximations of the Navier-Stokes equations by
projection schemes
Abstract: Thanks to their conceptional and computational simplicity projection schemes are very popular
and often rather efficient tools for the computational solution of the time dependent Navier-Stokes equations.
In this talk we present a posteriori error estimates for the (continuous in space) semi-discrete case, thus measuring the splitting error. We present estimates for the standard pressure correction scheme with backward
Euler discretization as well as for a BDF2 scheme in so called “rotation form”.
To the best of our knowledge these are the first results regarding a posteriori control for projection schemes.
T5. Soeren Bartels. Finite element approximation of functions of bounded variation
Abstract: Various phenomena involving free boundaries such as damage or plasticity require the description
of physical quantities with discontinuous functions. One approach to their mathematical modeling is based
on the space of functions of bounded variation which includes functions that are discontinuous and may
jump across lower dimensional subsets. Numerical methods for their approximate solution are often based
on regularizations which typically lead to restrictive conditions on discretization parameters. We try to
avoid such modifications and discuss the convergence of discretizations with different finite element spaces,
the iterative solution of the resulting finite-dimensional nonlinear systems of equations, and adaptive meshrefinement techniques based on rigorous a posteriori error estimates for a model problem related to image
processing. The application of the techniques to total variation flow, very singular diffusion processes, and
segmentation problems will be addressed.
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Part of this talk is based on joint work with Ricardo H. Nochetto (University of Maryland, USA) and Abner
J. Salgado (University of Maryland, USA).
T6. Andrea Bonito. Alternative Representation of Fractional Power of Self-Adjoint Elliptic Operators
Abstract: Taking advantage of the spectral properties of elliptic self-adjoint operators, we deduce a representation formula for their fractional powers. We show that in this context, fractional powers reduce to a
singular integral over the positive real numbers of a perturbation of the original operator.
Then, we deduce a novel numerical algorithm for the approximation of the fractional powers of such operators. A quadrature formula approximating the one dimensional singular integral is proposed while standard
finite element methods are advocated for the space discretization. The particularities of the proposed method
is that the quadrature points are distributed adequately to capture the (known) singularity and it reduces
to independent elliptic solves in space. The latter implies efficient scalability for parallel implementations.
We finally discuss optimal a-priori error estimates in terms of the number of degree of freedoms used for the
space discretization and the number of quadrature points.
T7. Luis Caffarelli. The homogenization of fronts and surfaces
Abstract: Most of the homogenization theory is develop for “bulk” quantities: densities, flows, etc.
In this lecture I will focus on some homogenization processes that concern surfaces, like phase transitions,
minimal surfaces or propagating fronts. In particular, I will describe in some detail geometric methods in
the context of a flame front model (this last part is joint work with Regis Monneau).
T8. Claudio Canuto. Adaptive high-order methods
Abstract: I will report on joint work with Ricardo and Marco Verani on adaptive algorithms for Fourier or
Legendre spectral methods. The nature of the approximation suggests a more aggressive attitude than for
finite-order methods; on the other hand, the complexity analysis must cope with sparsity classes in which the
best N-term approximation error decays faster than algebraically. This leads to some surprise. In the last
part of my talk, I will discuss the possibility of extending our framework of analysis to spectral-element/h-p
fem discretizations, where the dilemma ””refine or enrich”” poses new challenges.
T9. Zhiming Chen. An Adaptive Immersed Finite Element Method with Arbitrary Lagrangian-Eulerian
Scheme for Parabolic Equations in Variable Domains
Abstract: An adaptive immersed finite element method based on the a posteriori error estimate for solving
elliptic equations with non-homogeneous boundary condition in general Lipschitz domain is proposed. The
underlying finite element mesh need not to fit the boundary of the domain. Optimal a priori error estimate
of the proposed immersed finite element method is proved. The immersed finite element method is then used
to solve parabolic problems in time variable domains together with an arbitrary Lagrangian-Eulerian (ALE)
time discretization scheme. An a posteriori error estimate for the fully discrete immersed finite element
method is derived which can be used to adaptively update the time step sizes and finite element meshes at
each time step. Numerical results are reported to support the theoretical results. This is a joint work with
Zedong Wu and Yuanming Xiao.
T10. Long Chen. Multigrid methods for degenerate and singular elliptic equations
Abstract: In this talk, we will present fast multilevel methods for the approximate solution of the discrete
problems that arise from the discretization of fractional Laplacian. The fractional Laplacian is a nonlocal
operator. To localize it, we solve a Dirichlet to a Neumann-type operator via an extension problem. However,
this comes at the expense of incorporating one more dimension to the problem, thus motivates our study
of multilevel methods. We shall use the multilevel framework developed by Xu and Zikatanov and we
show nearly uniform convergence of a multilevel method for a class of general degenerate elliptic equations.
Because of the singularity of the solution, anisotropic elements in the extended variable are needed in order
to obtain quasi-optimal error estimates. For this reason, we also consider a multigrid method with a line
smoother and obtain nearly uniform convergence rates.
This is a joint work with Blanca Ayuso de Dios, Ricardo H. Nochetto, Enrique Ot’arola and Abner J. Salgado.
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T11. Bernardo Cockburn. Convergence for hybridizable discontinuous Galerkin methods
Abstract: We introduce the hybridizable discontinuous Galerkin methods for second-order elliptic equations
and place them in relation to already known discontinuous Galerkin methods, mixed methods and the
continuous Galerkin methods. Using this point of view, we review old and new work done on convergence
for discontinuous Galerkin methods. This is joint work with Ricardo Nochetto and Wujun Zhang.
T12. Ronald DeVore. Remarks on solving parametric elliptic problems
Abstract: We will discuss numerical strategies for solving parametric elliptic equations. Our emphasis will
be on two points of divergence from the typical strategies of Reduced Modeling. The one will be to allow
discontinuous diffusion coefficients. The second will be the possible incorporation of nonlinear methods into
the numerical procedure. Our results are very preliminary but may serve to focus future work.
T13. Georg Dolzmann. Modelling and simulation of vectorfields on surfaces
Abstract: We introduce a nonlinear model for the evolution of biomembranes driven by the gradient flow of
a novel elasticity functional describing the interaction of a director field on a membrane with its curvature.
In the linearized setting of a graph we present a practical finite element method (FEM), and prove a priori
estimates. We derive the relaxation dynamics for the nonlinear model on closed surfaces and introduce a
parametric FEM. We present numerical experiments which agree well with the expected behavior in model
situations. This is joint work with Soeren Bartels, Ricardo Nochetto and Alexander Raisch
T14. Willy Dörfler. A posteriori error estimation for indefinite Helmholtz problems
Abstract: We study the possibilities to get a posteriori error estimates for the solution of Helmholtz
problems that do not or only weakly depend on the wave number.
T15. Lucia Gastaldi. Finite elements for Immersed Boundary Method
Abstract: The aim of this talk is to discuss the performances of finite elements in the space discretization
of the Immersed Boundary Method. Immersed boundary solution is characterized by pressure discontinuities
at fluid structure interface. We analyze some popular Stokes elements such as Hood-Taylor and BercovierPirennau spaces together with some lowest order stabilizations. In particular, we investigate the local
mass conservation properties of the considered schemes and analyze new schemes with enhanced pressure
approximation, which guarantee a better local discretization of the divergence free constraint. Results show
that the enhanced pressure spaces are a significant cure for the well known “boundary leakage” affecting
IBM.
T16. Christian Kreuzer. Design and convergence analysis for an adaptive discretization of the heat equation
Abstract: We present an adaptive fully discrete space-time finite element method for the heat equation.
The algorithm is based on a classical adaptive time-stepping scheme supplemented by an additional control
of a potential energy increase of the discrete solution originating from coarsening of the spatial meshes. This
control allows to prove critical energy estimates in terms of given data from which one can derive an apriori
computable minimal time-step-size, which is sufficient for the required tolerance. The minimal step-size is
used by the algorithm and guarantees that the final time is reached in finitely many time-steps and within
a prescribed tolerance.
The minimal time-step-size has also a very positive effect in simulations. We present numerical experiments
that show a significant speedup compared to classical time-stepping schemes since too small time-steps are
avoided.
T17. Irene Kyza. Error control and adaptivity for linear Schrödinger equations in the semiclassical regime
Abstract: We derive optimal order a posteriori error bounds for a fully discrete Crank–Nicolson finite element scheme for linear Schrödinger equations. The derivation of the estimators is based on the reconstruction
technique; in particular, we introduce a novel elliptic reconstruction that leads to estimates which reflect
the physical properties of the equation. Our analysis also includes rough potentials. Using the obtained a
posteriori error estimators, we further develop and analyze an existing time-space adaptive algorithm, and
we apply it to the one-dimensional Schrödinger equation in the semiclassical regime. The adaptive algorithm
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reduces the computational cost drastically and provides efficient error control for the solution and the observables of the problem, especially for small values of the Planck constant.
This is a joint work with Th. Katsaounis.
T18. Omar Lakkis. Galerkin methods for fully nonlinear elliptic equations
Abstract: Fully nonlinear elliptic equations have been long left without a proper treatment by Galerkin
methods. I will review recent advances in efficient methods for the solution of fully nonlinear elliptic equations, such as Monge-Ampère and Pucci equations. I will focus on the technique of Hessian-recovery and
nonvariational Galerkin method.
This talk results from joint effort with Tristan Pryer (Kent, England).
T19. Jae-Hong Pyo. Error estimates for the second order semi-discrete stabilized Gauge–Uzawa method
for the Navier–Stokes equations
Abstract: The Gauge–Uzawa method [GUM], which is a projection type algorithm to solve the time depend
Navier–Stokes equations, has been constructed in [2] and enhanced in [3, 5] to apply to more complicated
problems. Even though GUM possesses many advantages theoretically and numerically, the studies on
GUM have been limited on the first order backward Euler scheme except normal mode error estimate in [4].
The goal of this paper is to research the 2nd order GUM. Because the classical 2nd order GUM which is
studied in [4] needs rather strong stability condition, we modify GUM to be unconditionally stable method
using BDF2 time marching. The stabilized GUM is equivalent to the rotational form of pressure correction
method and the errors are already estimated in [1] for the Stokes equations. In this paper, we will evaluate
errors of the stabilized GUM for the Navier–Stokes equations. We also prove that the stabilized GUM is an
unconditionally stable method for the Navier–Stokes equations. So we conclude that the rotational form of
pressure correction method in [1] is also unconditionally stable scheme and that the accuracy results in [1]
are valid for the Navier–Stokes equations.
[1] J.L. Guermond and J. Shen On the error estimates of rotational pressure-correction projec- tion methods,
Math. Comp., 73 (2004), 1719-1737.
[2] R.H. Nochetto and J.-H. Pyo, A finite element Gauge-Uzawa method. Part I : the Navier- Stokes
equations, SIAM J. Numer. Anal., 43, (2005), 1043–1068.
[3] R.H. Nochetto and J.-H. Pyo, A finite element Gauge-Uzawa method. Part II : Boussinesq Equations,
Math. Models Methods Appl. Sci., 16, (2006), 1599–1626.
[4] J.-H. Pyo and J. Shen, Normal Mode Analysis of Second-order Projection Methods for In- compressible
Flows, Discrete Contin. Dyn. Syst. Ser. B, 5, (2005), 817–840.
[5] J.-H. Pyo and J. Shen, Gauge Uzawa methods for incompressible flows with Variable Density, J. Comput.
Phys., 211, (2007), 181–197.
T20. Rodolfo Rodriguez. Numerical approximation of Beltrami fields
Abstract: Vector fields H satisfying curl H = λH, with λ being a scalar field, are called force-free fields.
This name arises from magnetohydrodynamics, since a magnetic field of this kind induces a vanishing Lorentz
force: F := J × B = curl H × (µH). In 1958 Woltjer [W] showed that the lowest state of magnetic energy
density within a closed system is attained when λ is spatially constant. In such a case H is called a linear
force-free field and its determination is naturally related with the spectral problem for the curl operator.
The eigenfunctions of this problem are known as free-decay fields and play an important role, for instance,
in the study of turbulence in plasma physics.
The spectral problem for the curl operator, curl H = λH, has a longstanding tradition in mathematical
physics. A large measure of the credit goes to Beltrami [B], who seems to be the first who considered this
problem in the context of fluid dynamics and electromagnetism. This is the reason why the corresponding
eigenfunctions are also called Beltrami fields. On bounded domains, the most natural boundary condition
for this problem is H · n = 0, which corresponds to a field confined within the domain. Analytical solutions
of this problem are only known under particular symmetry assumptions. The first one was obtained in 1957
by Chandrasekhar and Kendall [CK] in the context of astrophysical plasmas arising in modeling of the solar
crown.
More recently, some numerical methods have been introduced to compute force-free fields in domains without
symmetry assumptions [BA1,BA2]. In this work, we propose a variational formulation for the spectral
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problem for the curl operator which, after discretization, leads to a well-posed generalized eigenvalue problem.
We propose a method for its numerical solution based on Nédélec finite elements of arbitrary order. We
prove spectral convergence, optimal order error estimates and that the method is free of spurious-modes.
Finally we report some numerical experiments which confirm the theoretical results and allow us to assess
the performance of the method.
[B] E. Beltrami,Considerazioni idrodinamiche. Rend. Inst. Lombardo Acad. Sci. Let., vol. 22, pp.
122–131, (1889). (English translation: Considerations on hydrodynamics, Int. J. Fusion Energy, vol. 3, pp.
53–57, (1985).)
[BA1] T.Z. Boulmezaud, T. Amari,Approximation of linear force-free fields in bounded 3-D domains.
Math. Comp. Model., vol. 31, pp. 109–129, (2000).
[BA2] T.Z. Boulmezaud, T. Amari,A finite element method for computing nonlinear force-free fields.
Math. Comp. Model., vol. 34, pp. 903–920, (2001).
[CK] S. Chandrasekhar, P.C. Kendall,On force-free magnetic fields. Astrophys. J., vol. 126, pp.
457–460, (1957).
[N] J.C. Nédélec, Mixed finite elements in R3 . Numer. Math., vol. 35, pp. 315–341, (1980).
[W] L. Woltjer, A theorem on force-free magnetic fields. Prod. Natl. Acad. Sci. USA, vol. 44, pp.
489–491, (1958)
T21. Sandro Salsa. Free boundary problems with distributed sources: regularity results
Abstract: We describe new results on the regularity of the free boundary in two-phase problems governed
by second order elliptic equations with distributed sources. In particular, Lipschitz or suitable ””flat”” free
boundaries are smooth. Joint works with Daniela de Silva and Fausto Ferrari.
T22. Alfred Schmidt. FEM for phase transitions in welding processes
Abstract: We consider the simulation of solid-liquid phase transitions in the context of welding processes
and similar. The model includes heat transfer, melting and solidification, and free surface melt flow. Special
care is needed expecially where different free boundaries meet.
The talk presents joint work with Eberhard Bänsch and Jordi Paul (Erlangen) and Mischa Jahn and Andreas
Luttmann (Bremen).
T23. Kunibert G. Siebert. Adaptive finite elements for PDE constrained optimal control problems
Abstract: Many optimization processes in science and engineering lead to optimal control problems where
the sought state is a solution of a partial differential equation (PDE). Control and state may be subject
to further constraints. The complexity of such problems requires sophisticated techniques for an efficient
numerical approximation of the true solution. One particular method are adaptive finite element discretizations.
We report on ongoing research about control constrained optimal control problems. We give a summary
about recent findings concerning sensitivity analysis, a posteriori error control, and convergence of adaptive
finite elements.
This is joint work with Fernando D. Gaspoz (Stuttgart).
T24. Marco Verani. Hierarchical a posteriori error estimators for the mimetic discretization of elliptic
problems
Abstract: We present a posteriori error estimates of hierarchical type for the mimetic discretization of
elliptic problems. Under a saturation assumption, the global reliability and efficiency of the proposed a
posteriori estimators are proved. Several numerical experiments assess the actual performance of the local
error indicators in driving adaptive mesh refinement algorithms based on different marking strategies.
(Joint work with P.F. Antonietti, L. Beirao Da Veiga and C. Lovadina)
T25. Shawn Walker. A new mixed formulation for a sharp interface model of Stokes flow and moving
contact lines
Abstract: Two phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial
processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur
near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of
a Stokesian fluid with surface tension that interacts with a rigid substrate. The model is derived by an
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Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking level) and
allows for moving contact lines and contact angle hysteresis through a variational inequality. We prove the
well-posedness of the time semi-discrete and fully discrete (finite element) model and discuss error estimates.
Simulation movies will be presented to illustrate the method. We conclude with some discussion of a 3-D
version of the problem as well as future work on optimal control of these types of flows.
T26. Chensong Zhang. Adaptive Eulerian–Lagrangian Method for Convection-Diffusion Problems
Abstract: We consider the adaptive Eulerian–Lagrangian method (ELM) for linear convection-diffusion
problems. Unlike classical a posteriori error estimations, we estimate the temporal error along the characteristics and derive a new a posteriori error bound for ELM semi-discretization. With the help of this proposed
error bound, we are able to show the optimal convergence rate of ELM for solutions with minimal regularity.
Furthermore, by combining this error bound with a standard residual-type estimator for the spatial error, we
obtain a posteriori error estimators for a fully discrete scheme. We present numerical tests to demonstrate
the efficiency and robustness of our adaptive algorithm.
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