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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 allthe other, will enjoy this small-scale workshop focusing on recent advances in the Numerical Approximationof Partial Differential Equations in relation to Nochetto’s work.
Adaptive finite element methods are a fundamental numerical tool in science and engineering. They areknown to outperform classical finite element in practice and deliver optimal convergence rates when the lattercannot. The aim of this workshop is to stimulate a fruitful discussion regarding adaptivity, error control andconvergence 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 XXIVMaggio, 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, OmarLakkis, Pedro Morin, Andreas Veeser, Marco Verani, Claudio Verdi, Chen-Song Zhang.
The OC acknowledges Giuseppe Savaré’s very important contribution and suggestions. Special thanks goto Marie Bonito for the design of the poster.
The Scientific Committee (SC) is formed by: Claudio Canuto, Zhi-Ming Chen, Ronald DeVore, VivetteGirault, 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 acknowl-edged.
Hopefully this booklet contains all the important information for a smooth running of events, but it isvery likely that something was missed out. In that case, if you need assistance, please refer directly to thatmember of the OC that is closest to you.
Contents
1. Happy birthday to Ricardo and welcome to Gargnano 22. Travel information 33. Maps 54. Workshop information 65. Participants 76. Schedule 117. Talk data 12
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2. Travel information
The following information on how to reach Gargnano has been copy-pasted fromhttp://www.gargnanosulgarda.it/tourist-service-lakegarda/how-to-reach-gargnano.htmlwhere 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 frontaround the corner on the left, there’s the Bus Stand to the Lake (not the round one on the right), whoselast 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 stationand get a bus to Gargnano by another line. Another option is to join any location on the lake and get aboat to Gargnano
From Rovereto. Get to Riva del Garda with this bus line. Once in Riva, get a bus to Desenzano, which stopsin 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 delGarda direction and after average 13 km you will find Bogliaco, Villa and last, Gargnano Center. Take theonly road entering the center.
From Venice. Take Highway A4 Venezia-Milano, exit at Desenzano del Garda and go to Salò.Then take Rivadel 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 findGargnano. U turn to enter in the center.
2.3. From Malpensa (Milan) airport. Bus/Train: just out of the terminal take the Malpensa Shuttle toMilano Centrale RLW Station, catch the first train to Brescia, once out Brescia RLW Station, 200 metersfront 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 directionRiva del Garda. After around 15 km, you find Bogliaco, then Villa and then Gargnano Centro. Take theonly road entering the center.
2.4. From Linate (Milan) airport. Bus/Train: just out of the terminal take any public mean to joinMilano Centrale RLW Station, catch the first train to Brescia, once out Brescia RLW Station, 200 metersfront 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 thedirection 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 RLWStation and from there, take the train to Brescia, once out Brescia RLW Station, 200 meters front left, takethe 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 thedirection 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 boatof Navigarda and join Gargnano.
Bus/Train: Catch any public mean to join Verona RLW Station and from there, take the train to Desen-zano, and then the bus to Gargnano or join Brescia RLW Station.Once out, 200 meters front left, take thebus 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 directionRiva del Garda. After around 15 km, you find Bogliaco, then Villa and then Gargnano Centro. Take theonly road entering the center.
2.7. From Montichiari “D’Annunzio” airport. Boat: join any locality of the lake, from there you cantake 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 toSalò. Then take the direction Riva del Garda. After around 15 km, you find Bogliaco, then Villa and thenGargnano 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 thenthe 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 directionRiva del Garda. After around 15 km, you find Bogliaco, then Villa and then Gargnano Centro. Take theonly road entering the center.
2.9. From Rovereto railway station. Bus: take the bus Rovereto-Riva del Garda from here, catch thebus 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 ofthe Lake Garda, in the Italian Province ofBrescia, in the middle of the Regional Park“Alto Garda”. It has a very interesting his-toric background, including the history ofPalazzo Feltrinelli where the workshop istaking place. Lake Garda being a primetourist destination there are plenty of extra-academic activities for you to partake infrom windsurfing to hiking in the park.
Gargnano can be reached by road and byboat. The closest major railway stations areDesenzano (south-east) and Brescia (south-west). It is also possible to reach it by boat.Check the travel information for more.
Gargnano is quite small and cosy. Fromthe bus station to Palazzo Feltrinelli is 5 aminutes 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 re-member the non-invited (’not yellow’) that the payments for the meals and the accommodation in PalazzoFeltrinelli and Casa Bertolini has to be done in cash. The rooms of Palazzo Feltrinelli and Casa Bertolinihave to be left on the departure day by 9:00 (otherwise another day will be charged). People with accom-modation at Casa Bertolini (50 m from the Palazzo) have to register at Palazzo Feltrinelli in order to getthe key of their room.
4.3. Accommodation. You should by now have all the details of your accommodation which, barringpersonal 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. Pleasebe on time to avoid missing out. According to the house rules, you must be registered for each lunch ordinner you are planning to have. Should the information you received be different, please contact a memberof 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 youwill need to present a valid ID (e.g., passport) a copy of which has to be made by the staff (Italian law). InCasa Bertolini, there is unfortunately, no direct wireless access. In the Hotel Meandro, you should ask thehotel staff.
4.7. Audiovisual aids. All lectures and talks will take place in the “Aula magna”. We are planning toarrange a single computer-projector set-up, so speakers should have a PDF-ready talk (if you have morecomplicated presentations, e.g., movies, non-PDF format, etc. please let the organizers know as early as youcan).
4.8. Other questions? Ask a member of the OC.
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5. Participants
Georgios Akrivis: [email protected] of IoanninaDepartment of Computer ScienceIoannina, Greece
Harbir Antil: [email protected] Mason UniversityDepartment of Mathematical SciencesFairfax, Virginia USA
Doug Arnold: [email protected] of MinnesotaSchool of MathematicsMinneapolis, Minnesota USA
Ayuso de Dios Blanca: [email protected] de Recerca MatemàticaBellaterraBarcelona, Catalunya, Spain
Eberhard Bänsch: [email protected] of ErlangenDepartment MathematikErlangen, Germany
Soeren Bartels: [email protected] of FreiburgDepartment Angewandte MathematikFreiburg, Germany
Stefano Berrone: [email protected] di TorinoDipartimento di Scienze MatematicheTorino, Italy
Daniele Boffi: [email protected] of PaviaDipartimento di MatematicaPavia, Italy
Andrea Bonito: [email protected] A & M UniversityDepartment of MathematicsCollege Station, Texas USA
Andrea Bressan: [email protected]à di PaviaDipartimento di MatematicaPavia, Italy
Luis Caffarelli: [email protected] of Texas AustinDepartment of MathematicsAustin, Texas USA
Claudio Canuto: [email protected] di TorinoDipartimento di Scienze MatematicheTorino, Italy
Lara Antonella Charawi: [email protected]à degli Studi di Milano
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Dipartimento di MatematicaMilano, Italy
Zhiming Chen: [email protected] Academy of SciencesAcademy of Mathematics and Systems ScienceBeijing, China
Long Chen: [email protected] of California at IrvineDepartment of MathematicsIrvine, California USA
Bernardo Cockburn: [email protected] of MinnesotaSchool of MathematicsMinneapolis, Minnesota USA
Piero Colli Franzone: [email protected]à di PaviaDipartimento di MatematicaPavia, Italia
Ronald DeVore: [email protected] A & M UniversityDepartment of MathematicsCollege Station, Texas USA
Georg Dolzmann: [email protected]ät RegensburgFakultät für MathematikRegensburg, Germany
Willy Dörfler: [email protected] Institute of TechnologyDepartment of MathematicsKarlsruhe, Germany
Fernando Gaspoz: [email protected]ät StuttgartInstituts für Angewandte Analysis und Numerische SimulationStuttgart, Germany
Lucia Gastaldi: [email protected] of BresciaDipartimento di MatematicaBrescia, Italy
Fotini Karakatsani: [email protected] of StrathclydeMathematics and StatisticsGlasgow, Scotland UK
Theodoros Katsaounis: [email protected] of CreteDepartment of Applied MathematicsHeraklion, Greece
Christian Kreuzer: [email protected] Universität BochumFakultät für MathematikBochum, Germany
Irene Kyza: [email protected] of Dundee
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Division of MathematicsDundee, Scotland UK
Omar Lakkis: [email protected] of SussexDepartment of MathematicsBrighton, England UK
Charalambos Makridakis: [email protected] of CreteDepartment of Applied MathematicsHeraklion, Greece
Giovanni Migliorati: [email protected]École Polythechnique Fédérale de LausanneMATHICSELausanne, Switzerland
Pedro Morin: [email protected] Nacional del LitoralIMAL, CONICETSanta Fe, Argentina
Ricardo Nochetto: [email protected] of MarylandDepartment of Mathematics & IPSTCollege Park, Maryland USA
Maurizio Paolini: [email protected]’ Cattolica BresciaDipartimento di Matematica e FisicaBrescia, Italy
Sebastian Pauletti: [email protected] Nazionale delle Ricerche (CNR)IMATI ’Enrico Magenes’Pavia, Italy
Ilaria Perugia: [email protected]à di PaviaDipartimento di MatematicaPavia, Italy
Paola Pietra: [email protected] Nazionale delle Ricerche (CNR)IMATI ’Enrico Magenes’Pavia, Italy
Jae-Hong Pyo: [email protected] National UniversityDepartment of MathematicsChuncheon, Korea
Rodolfo Rodriguez: [email protected] de ConcepcionDepartamento de Ingenieria MatematicaConcepcion, Chile
Sandro Salsa: [email protected] di MilanoDipartimento di Matematica ””Francesco Brioschi””Milano, Italy
Giuseppe Savaré: [email protected] of Pavia
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Department of MathematicsPavia, Italy
Alfred Schmidt: [email protected]ät BremenZentrum für TechnomathematikBremen, Germany
Natasha Sharma: [email protected] of HeidelbergIWRHeidelberg, Germany
Kunibert G. Siebert: [email protected]ät StuttgartInstituts für Angewandte Analysis und Numerische SimulationStuttgart, Germany
Francesca Tantardini: [email protected]à di MilanoDipartimento di MatematicaMilano, Italy
Andreas Veeser: [email protected]à di MilanoDipartimento di MatematicaMilano, Italy
Marco Verani: [email protected] di MilanoDipartimento di MatematicaMilano, Italy
Claudio Verdi: [email protected]à di MilanoDipartimento di MatematicaMilano, Italy
Shawn Walker: [email protected] State UniversityDepartment of MathematicsBaton Rouge, Louisiana USA
Chensong Zhang: [email protected] Academy of SciencesAcademy of Mathematics and System SciencesBeijing, China
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6. ScheduleTuesday, 19 March 2013
15:00 to 19:30 registration19:30 to 21:30 dinner
Wednesday, 20 March 201308:45 to 09:00 welcome09:00 to 09:50 keynote talk by Ronald DeVore (T12)10:00 to 10:30 invited talk by Georgios Akrivis (T1)10:35 to 11:00 coffee11:00 to 11:30 invited talk by Eberhard Bänsch (T4)11:35 to 12:05 invited talk by Zhiming Chen (T9)12:10 to 12:40 invited talk by Irene Kyza (T17)12:45 to 13:45 lunch15:30 to 16:00 invited talk by Andrea Bonito (T6)16:05 to 16:35 invited talk by Marco Verani (T24)16:40 to 17:05 coffee17:05 to 17:55 keynote talk by Bernardo Cockburn (T11)18:05 to 18:35 invited talk by Christian Kreuzer (T16)18:40 to 19:30 social19:30 to 21:30 dinner
Thursday, 21 March 201309:00 to 09:50 keynote talk by Luis Caffarelli (T7)10:00 to 10:30 invited talk by Rodolfo Rodriguez (T20)10:35 to 11:00 coffee11:00 to 11:30 invited talk by Lucia Gastaldi (T15)11:35 to 12:05 invited talk by Harbir Antil (T2)12:10 to 12:40 invited talk by Sören Bartels (T5)12:45 to 13:45 lunch15:30 to 16:00 invited talk by Long Chen (T10)16:05 to 16:35 invited talk by Jae-Hong Pyo (T19)16:40 to 17:05 coffee17:05 to 17:55 keynote talk by Douglas Arnold (T3)18:05 to 18:35 invited talk by Shawn Walker (T25)18:40 to 19:30 social19:30 to 00:00 conference dinner followed by party
Friday, 22 March 201309:00 to 09:50 keynote talk by Sandro Salsa (T21)10:00 to 10:30 invited talk by Alfred Schmidt (T22)10:35 to 11:00 coffee11:00 to 11:30 invited talk by Chensong Zhang (T26)11:35 to 12:05 invited talk by Kunibert G. Siebert (T23)12:35 to 13:35 lunch14:10 to 14:40 invited talk by Georg Dolzmann (T13)14:45 to 15:15 invited talk by Omar Lakkis (T18)15:20 to 15:45 coffee15:45 to 16:15 invited talk by Willy Dörfler (T14)16:20 to 16:50 invited talk by Claudio Canuto (T8)18:30 to 20:30 dinner
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7. Talk data
T1. Georgios Akrivis. Implicit-explicit multistep methods for a class of nonlinear parabolic equationsAbstract: 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 willdiscuss consistency and stability of the schemes, under certain conditions, and will derive optimal ordererror estimates. The stability assumptions can be relaxed in the case of first and second order schemes. Thediscretization of nonlinear convection-diffusion equations by implicit-explicit multistep schemes will also bebriefly discussed.
T2. Harbir Antil. A Stokes free boundary problem with surface tension effectsAbstract: We consider a Stokes free boundary problem with surface tension effects in variational form. Thismodel is an extension of the coupled system proposed by P. Saavedra and L. R. Scott, where they considera Laplace equation in the bulk with Young-Laplace equation on the free boundary to account for surfacetension. The two main difficulties for the Stokes free boundary problem are: the vector curvature on theinterface, which causes problem to write a variational form of the free boundary problem and the existence ofsolution to Stokes equations with Navier-slip boundary conditions for W 1+1/p′
p domains (minimal regularity).We will demonstrate the existence of solution to Stokes equations with Navier-slip boundary conditions usinga perturbation argument for the bended half space followed by standard localization technique. The W
1+1/p′
p
regularity of the interface allows us to write the variational form for the entire free boundary problem, weconclude with the well-posedness of this system using a fixed point iteration.
T3. Doug Arnold. The periodic table of finite elementsAbstract: Finite element methodology, reinforced by deep mathematical analysis, provides one of the mostimportant and powerful toolsets for numerical simulation. Over the past forty years a bewildering varietyof different finite element spaces have been invented to meet the demands of many different problems. Therelationship between these finite elements has often not been clear, and the techniques developed to analyzethem can seem like a collection of ad hoc tricks. The finite element exterior calculus, developed over thelast 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 topreviously intractable problems. In this talk, we will discuss the big picture that emerges, providing a sortof periodic table of finite element methods.
T4. Eberhard Bänsch. A posteriori error estimates for approximations of the Navier-Stokes equations byprojection schemesAbstract: Thanks to their conceptional and computational simplicity projection schemes are very popularand 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 mea-suring the splitting error. We present estimates for the standard pressure correction scheme with backwardEuler 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 variationAbstract: Various phenomena involving free boundaries such as damage or plasticity require the descriptionof physical quantities with discontinuous functions. One approach to their mathematical modeling is basedon the space of functions of bounded variation which includes functions that are discontinuous and mayjump across lower dimensional subsets. Numerical methods for their approximate solution are often basedon regularizations which typically lead to restrictive conditions on discretization parameters. We try toavoid 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 mesh-refinement techniques based on rigorous a posteriori error estimates for a model problem related to imageprocessing. The application of the techniques to total variation flow, very singular diffusion processes, andsegmentation 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 AbnerJ. Salgado (University of Maryland, USA).
T6. Andrea Bonito. Alternative Representation of Fractional Power of Self-Adjoint Elliptic OperatorsAbstract: Taking advantage of the spectral properties of elliptic self-adjoint operators, we deduce a repre-sentation formula for their fractional powers. We show that in this context, fractional powers reduce to asingular 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 opera-tors. A quadrature formula approximating the one dimensional singular integral is proposed while standardfinite element methods are advocated for the space discretization. The particularities of the proposed methodis that the quadrature points are distributed adequately to capture the (known) singularity and it reducesto 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 thespace discretization and the number of quadrature points.
T7. Luis Caffarelli. The homogenization of fronts and surfacesAbstract: 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 inthe context of a flame front model (this last part is joint work with Regis Monneau).
T8. Claudio Canuto. Adaptive high-order methodsAbstract: I will report on joint work with Ricardo and Marco Verani on adaptive algorithms for Fourier orLegendre spectral methods. The nature of the approximation suggests a more aggressive attitude than forfinite-order methods; on the other hand, the complexity analysis must cope with sparsity classes in which thebest N-term approximation error decays faster than algebraically. This leads to some surprise. In the lastpart of my talk, I will discuss the possibility of extending our framework of analysis to spectral-element/h-pfem discretizations, where the dilemma ””refine or enrich”” poses new challenges.
T9. Zhiming Chen. An Adaptive Immersed Finite Element Method with Arbitrary Lagrangian-EulerianScheme for Parabolic Equations in Variable DomainsAbstract: An adaptive immersed finite element method based on the a posteriori error estimate for solvingelliptic equations with non-homogeneous boundary condition in general Lipschitz domain is proposed. Theunderlying finite element mesh need not to fit the boundary of the domain. Optimal a priori error estimateof the proposed immersed finite element method is proved. The immersed finite element method is then usedto 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 elementmethod is derived which can be used to adaptively update the time step sizes and finite element meshes ateach time step. Numerical results are reported to support the theoretical results. This is a joint work withZedong Wu and Yuanming Xiao.
T10. Long Chen. Multigrid methods for degenerate and singular elliptic equationsAbstract: In this talk, we will present fast multilevel methods for the approximate solution of the discreteproblems that arise from the discretization of fractional Laplacian. The fractional Laplacian is a nonlocaloperator. 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 studyof multilevel methods. We shall use the multilevel framework developed by Xu and Zikatanov and weshow 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 orderto obtain quasi-optimal error estimates. For this reason, we also consider a multigrid method with a linesmoother 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 methodsAbstract: We introduce the hybridizable discontinuous Galerkin methods for second-order elliptic equationsand place them in relation to already known discontinuous Galerkin methods, mixed methods and thecontinuous Galerkin methods. Using this point of view, we review old and new work done on convergencefor discontinuous Galerkin methods. This is joint work with Ricardo Nochetto and Wujun Zhang.
T12. Ronald DeVore. Remarks on solving parametric elliptic problemsAbstract: We will discuss numerical strategies for solving parametric elliptic equations. Our emphasis willbe on two points of divergence from the typical strategies of Reduced Modeling. The one will be to allowdiscontinuous diffusion coefficients. The second will be the possible incorporation of nonlinear methods intothe numerical procedure. Our results are very preliminary but may serve to focus future work.
T13. Georg Dolzmann. Modelling and simulation of vectorfields on surfacesAbstract: We introduce a nonlinear model for the evolution of biomembranes driven by the gradient flow ofa 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 prioriestimates. We derive the relaxation dynamics for the nonlinear model on closed surfaces and introduce aparametric FEM. We present numerical experiments which agree well with the expected behavior in modelsituations. This is joint work with Soeren Bartels, Ricardo Nochetto and Alexander Raisch
T14. Willy Dörfler. A posteriori error estimation for indefinite Helmholtz problemsAbstract: We study the possibilities to get a posteriori error estimates for the solution of Helmholtzproblems that do not or only weakly depend on the wave number.
T15. Lucia Gastaldi. Finite elements for Immersed Boundary MethodAbstract: The aim of this talk is to discuss the performances of finite elements in the space discretizationof the Immersed Boundary Method. Immersed boundary solution is characterized by pressure discontinuitiesat fluid structure interface. We analyze some popular Stokes elements such as Hood-Taylor and Bercovier-Pirennau spaces together with some lowest order stabilizations. In particular, we investigate the localmass conservation properties of the considered schemes and analyze new schemes with enhanced pressureapproximation, which guarantee a better local discretization of the divergence free constraint. Results showthat the enhanced pressure spaces are a significant cure for the well known “boundary leakage” affectingIBM.
T16. Christian Kreuzer. Design and convergence analysis for an adaptive discretization of the heat equa-tionAbstract: 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 controlof a potential energy increase of the discrete solution originating from coarsening of the spatial meshes. Thiscontrol allows to prove critical energy estimates in terms of given data from which one can derive an aprioricomputable minimal time-step-size, which is sufficient for the required tolerance. The minimal step-size isused by the algorithm and guarantees that the final time is reached in finitely many time-steps and withina prescribed tolerance.The minimal time-step-size has also a very positive effect in simulations. We present numerical experimentsthat show a significant speedup compared to classical time-stepping schemes since too small time-steps areavoided.
T17. Irene Kyza. Error control and adaptivity for linear Schrödinger equations in the semiclassical regimeAbstract: We derive optimal order a posteriori error bounds for a fully discrete Crank–Nicolson finite ele-ment scheme for linear Schrödinger equations. The derivation of the estimators is based on the reconstructiontechnique; in particular, we introduce a novel elliptic reconstruction that leads to estimates which reflectthe physical properties of the equation. Our analysis also includes rough potentials. Using the obtained aposteriori error estimators, we further develop and analyze an existing time-space adaptive algorithm, andwe 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 ob-servables 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 equationsAbstract: Fully nonlinear elliptic equations have been long left without a proper treatment by Galerkinmethods. I will review recent advances in efficient methods for the solution of fully nonlinear elliptic equa-tions, such as Monge-Ampère and Pucci equations. I will focus on the technique of Hessian-recovery andnonvariational 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 methodfor the Navier–Stokes equationsAbstract: The Gauge–Uzawa method [GUM], which is a projection type algorithm to solve the time dependNavier–Stokes equations, has been constructed in [2] and enhanced in [3, 5] to apply to more complicatedproblems. Even though GUM possesses many advantages theoretically and numerically, the studies onGUM 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 isstudied in [4] needs rather strong stability condition, we modify GUM to be unconditionally stable methodusing BDF2 time marching. The stabilized GUM is equivalent to the rotational form of pressure correctionmethod and the errors are already estimated in [1] for the Stokes equations. In this paper, we will evaluateerrors of the stabilized GUM for the Navier–Stokes equations. We also prove that the stabilized GUM is anunconditionally stable method for the Navier–Stokes equations. So we conclude that the rotational form ofpressure 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- Stokesequations, 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- compressibleFlows, 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 fieldsAbstract: Vector fields H satisfying curlH = 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 Lorentzforce: F := J ×B = curlH × (µH). In 1958 Woltjer [W] showed that the lowest state of magnetic energydensity within a closed system is attained when λ is spatially constant. In such a case H is called a linearforce-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, curlH = λH, has a longstanding tradition in mathematicalphysics. A large measure of the credit goes to Beltrami [B], who seems to be the first who considered thisproblem in the context of fluid dynamics and electromagnetism. This is the reason why the correspondingeigenfunctions are also called Beltrami fields. On bounded domains, the most natural boundary conditionfor this problem is H ·n = 0, which corresponds to a field confined within the domain. Analytical solutionsof this problem are only known under particular symmetry assumptions. The first one was obtained in 1957by Chandrasekhar and Kendall [CK] in the context of astrophysical plasmas arising in modeling of the solarcrown.More recently, some numerical methods have been introduced to compute force-free fields in domains withoutsymmetry 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. Weprove 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 assessthe 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 resultsAbstract: We describe new results on the regularity of the free boundary in two-phase problems governedby second order elliptic equations with distributed sources. In particular, Lipschitz or suitable ””flat”” freeboundaries are smooth. Joint works with Daniela de Silva and Fausto Ferrari.
T22. Alfred Schmidt. FEM for phase transitions in welding processesAbstract: We consider the simulation of solid-liquid phase transitions in the context of welding processesand similar. The model includes heat transfer, melting and solidification, and free surface melt flow. Specialcare is needed expecially where different free boundaries meet.The talk presents joint work with Eberhard Bänsch and Jordi Paul (Erlangen) and Mischa Jahn and AndreasLuttmann (Bremen).
T23. Kunibert G. Siebert. Adaptive finite elements for PDE constrained optimal control problemsAbstract: Many optimization processes in science and engineering lead to optimal control problems wherethe sought state is a solution of a partial differential equation (PDE). Control and state may be subjectto further constraints. The complexity of such problems requires sophisticated techniques for an efficientnumerical approximation of the true solution. One particular method are adaptive finite element discretiza-tions.We report on ongoing research about control constrained optimal control problems. We give a summaryabout recent findings concerning sensitivity analysis, a posteriori error control, and convergence of adaptivefinite elements.This is joint work with Fernando D. Gaspoz (Stuttgart).
T24. Marco Verani. Hierarchical a posteriori error estimators for the mimetic discretization of ellipticproblemsAbstract: We present a posteriori error estimates of hierarchical type for the mimetic discretization ofelliptic problems. Under a saturation assumption, the global reliability and efficiency of the proposed aposteriori estimators are proved. Several numerical experiments assess the actual performance of the localerror 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 movingcontact linesAbstract: Two phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrialprocesses, such as micro-fluidics and coating flows. These flows include additional physical effects that occurnear moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation ofa 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) andallows for moving contact lines and contact angle hysteresis through a variational inequality. We prove thewell-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-Dversion 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 ProblemsAbstract: We consider the adaptive Eulerian–Lagrangian method (ELM) for linear convection-diffusionproblems. Unlike classical a posteriori error estimations, we estimate the temporal error along the character-istics and derive a new a posteriori error bound for ELM semi-discretization. With the help of this proposederror 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, weobtain a posteriori error estimators for a fully discrete scheme. We present numerical tests to demonstratethe efficiency and robustness of our adaptive algorithm.
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