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## Kolloquium der Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen

Verantwortliche Dozenten: | Alle Professoren der Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen |
---|---|

Koordination: | Prof. Dr. Christian Schröder |

Termine: | Di 16-18 Uhr in MA 313 und nach Vereinbarung |

Inhalt: | Vorträge von Gästen und Mitarbeitern zu aktuellen Forschungsthemen |

## Beschreibung

Das Kolloquium der Arbeitsgruppe "Modellierung, Numerik, Differentialgleichungen" im Institut der Mathematik ist ein Kolloquium klassischer Art. Es wird also von einem breiten Kreis der Professoren und Mitarbeiter aus allen zugehörigen Lehrstühlen, insbesondere Angewandte Funktionalanalysis, Numerische Lineare Algebra, und Partielle Differentialgleichungen, besucht. Auch Studierende nach dem Bachelorabschluss zählen schon zu den Teilnehmern unseres Kolloquiums.

Aus diesen Gründen freuen wir uns insbesondere über Vorträge, die auf einen nicht spezialisierten Hörerkreis zugeschnitten sind und auch von Studierenden nach dem Bachelorabschuss bereits mit Profit gehört werden können.

Datum date | Zeit time | Raum room | Vortragende(r) speaker | Titel title | Einladender invited by |
---|---|---|---|---|---|

Di 10.04.12 | 16:15 | MA 313 | Cesar Palencia (U Valladolid) | Runge-Kutta convolution quadrature for abstract, linear, homogeneous Volterra equations (Abstract) | E. Emmrich |

Di 17.04.12 | 16:15 | MA 313 | Massimo Fornasier (TU Muenchen) | Analysis and Simulation of Particle Systems and Kinetic Equations Modeling Interacting Agents in High Dimension (Abstract) | G. Kutyniok |

Di 24.04.12 | 16:15 | MA 313 | Moritz Kaßmann (U Bielefeld) | Regularity theory for parabolic nonlocal operators (Abstract) | E. Emmrich |

Di 8.05.12 | 16:15 | MA 313 | Vladimir Sergeichuk (Nat Akad Ukraine, Kiev) | Classification problems for systems of forms and linear mappings (Abstract) | C. Mehl V. Mehrmann |

Di 15.05.12 | 16:15 | MA 313 | Dozenten der AG ModNumDiff | Lehrbesprechung AG ModNumDiff | I. Yousept |

Di 22.05.12 | 16:15 | MA 313 | Janosch Rieger (U Frankfurt a.M.) | Galerkin Finite Elements for Partial Differential Inclusions (Abstract) | E. Emmrich |

Di 29.05.12 | 16:15 | MA 313 | Kasso Okoudjou (U of Maryland) | Probabilistic frames (Abstract) | G. Kutyniok J. Vybiral |

Di 5.06.12 | 16:15 | MA 313 | Boris Andreianov (U Besançon) | Existence and finite volume approximation for a cross-diffusion system (Abstract) | E. Emmrich |

followed by | MA 313 | Zouwei Shen (National U Singapore) | MRA based wavelet frame and applications (Abstract) | G. Kutyniok | |

Di 12.06.12 | 16:15 | MA 313 | Hamdullah Yücel (METU Ankara) | Adaptive Discontinuous Galerkin Methods for Advection Dominated Optimal Control Problems (Abstract) | A. Schiela |

Di 19.06.12 | 16:15 | MA 313 | Daniel Peterseim (HU Berlin) | Finite Element Computational Homogenization of Multiscale Elliptic Problems (Abstract) | H. Yserentant K. Schmidt |

Mi 20.06.12 | 18:00 | MA 005 | Emmanuel Candes (Stanford U) | Compressive Sensing (Abstract) | G. Kutyniok |

Di 26.06.12 | 16:15 | MA 313 | Chun Liu (Penn State U) | Energetic Variational Approaches in Complex Fluids (Abstract) | B. Wagner |

Di 3.07.12 | 16:15 | MA 313 | Philipp Grohs (ETH Zuerich) | Algorithmic Treatment of Nonlinear Data (Abstract) | G. Kutyniok |

followed by | MA 313 | Guy Vallet (U Pau) | On stochastic conservation laws (Abstract) | E. Emmrich | |

Di 10.07.12 | 16:15 | MA 313 | Stephan Dahlke (U Marburg) | The Continuous Shearlet Transform in Arbitrary Space Dimension: General Setting, Shearlet Coorbit Spaces, Traces, and Embeddings (Abstract) | G. Kutyniok |

Do 9.08.12 | 16:15 | MA 313 | Martin Gugat (U Erlangen-Nürnberg) | Boundary Feedback Control of PDE-Systems: Stability Analysis with Lyapunov Functions (Abstract) | F. Tröltzsch |

Di 21.08.12 | 16:15 | MA 313 | Tadele Mengesha (Pennsylvania State U) | Mathematical Analysis of the linear peridynamic model (Abstract) | E. Emmrich |

followed by | MA 313 | Dorothee Knees (WIAS Berlin) | On a vanishing viscosity approach in damage mechanics (Abstract) | E. Emmrich | |

Mi 22.08.12 | 16:15 | MA 551 | Inderjit Dhillon (U Texas) | Orthogonal Matching Pursuit with Replacement (Abstract) | V. Mehrmann |

Di 28.08.12 | 16:15 | MA 313 | Julian Diaz (U Pau, INRIA Bordeaux) | Stability analysis of the Interior Penalty Discontinuous Galerkin method for the wave equation (Abstract) | K. Schmidt |

Fr 31.08.12 | 9:30 | MA 415 | Sebastien Tordeux (U Pau, INRIA Bordeaux) | Perforated and multiperforated plates in linear acoustic (Abstract) | K. Schmidt |

### Cesar Palencia (U Valladolid)

**Runge-Kutta convolution quadrature for abstract, linear, homogeneous Volterra equations**

Dienstag, den 10.04.2012, 16.15 Uhr in MA 313

Abstract:

Recent results concerning Runge-Kutta based convolution quadrature methods for abstract, well posed, linear, and homogeneous Volterra equations show a general representation of the numerical solution in terms of the continuous one. The interest of such a representation goes beyond the error and stability analysis, since it explains that the numerical solution, under suitable methods, inherits some important qualitative properties, such as positivity or contractivity, of the exact solution. In this talk we will summarize this theory and use it for the construction of fully discrete, transparent boundary conditions, for the heat and Schroedinger equation, with good qualitative properties.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

### Massimo Fornasier (TU Muenchen)

**Analysis and Simulation of Particle Systems and Kinetic Equations Modeling Interacting Agents in High Dimension**

Dienstag, den 17.04.2012, 16.15 Uhr in MA 313

Abstract:

We are addressing the analysis and the tractable simulation of dynamical systems which are modeling the behavior in a social context of a large number N of complex interacting agents described by a large amount of parameter (high-dimension).We are facing the following fundamental challenges and by describing them in detail we are also pointing to new research directions:

- Random projections and recovery for high-dimensional dynamical systems: we shall explore how concepts of data compression via Johnson-Lindenstrauss random embeddings onto lower-dimensional spaces can be applied for tractable description and simulation of complex dynamical interactions. As a fundamental subtask for the recovery of high-dimensional trajectories from low-dimensional simulated ones, we will address the efficient recovery of point clouds defined on embedded manifolds from random projections.

- Mean field equations: for the limit of the number N of agents to infinity, we shall further explore how the concepts of compression can be generalized to work for associated mean field equations.

- Approximating functions in high-dimension: differently from purely physical problems, in the real life the social forces which are ruling the dynamics are actually not known. Hence we will address the problem of automatic learning from collected data the fundamental functions governing the dynamics.

- Sparse optimal control in high-dimension and mean field optimal control: while self-organization of such dynamical systems has been so far a mainstream, we will focus on their sparse optimal control, modeling best policies. We will investigate L1-minimization to design sparse optimal controls.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

### Moritz Kaßmann (U Bielefeld)

**Regularity theory for parabolic nonlocal operators**

Dienstag, den 24.04.2012, 16.15 Uhr in MA 313

Abstract:

We extend Moser's works from 1964, 1967 and 1971 to nonlocal problems, i.e. we prove Hölder-regularity and a Harnack-type inequality for weak solutions to parabolic equations with integro-differential operators of fractional order. Assumptions on the kernel are discussed in detail and several examples are provided. This work is joint with M. Felsinger.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

### Vladimir Sergeichuk (National Academy Ukraine, Kiev)

**Classification problems for systems of forms and linear mappings**

Dienstag, den 8.05.2012, 16.15 Uhr in MA 313

Abstract:

A method that reduces the problem of classifying systems of bilinear/sesquilinear forms and linear mappings to the problem of classifying systems of linear mappings is presented.

### Janosch Rieger (U Frankfurt a.M.)

**Galerkin Finite Elements for Partial Differential Inclusions**

Dienstag, den 22.05.2012, 16.15 Uhr in MA 313

Abstract:

The partial differential inclusions considered in this talk arise in a natural way from the study of constrained control problems and deterministic uncertainty. In this setting, the solution set contains all solutions induced by admissible controls or perturbations. We currently try to develop the necessary analytical background and numerical methods for an efficient approximation of this solution set.

First experiments in the linear elliptic case show that it is difficult to obtain good results by discretizing the multivalued right-hand side $F$. It is much better to project the differential inclusion to some finite-dimensional space and approximate the solution set of the resulting algebraic inclusion.

The semi-linear elliptic case is much more involved. Set-valued Nemytskii operators have to be considered for the projection of the inclusion to a finite element space. In order to guarantee uniform convergence of the Galerkin solution sets, the so-called \emph{relaxed one-sided Lipschitz property} is imposed on the right-hand side $F$, which is a generalization of the classical OSL property. Under this assumption, it is also possible to discretize and approximate the unknown Galerkin solution sets.

Work on semi-linear parabolic inclusions is still in progress, but almost completed. The convergence of the Galerkin solution sets has been established. In this case, the Galerkin inclusions are ordinary differential inclusions, and their reachable sets can (at least theoretically) be computed by availabe numerical methods.

### Kasso Okoudjou (U of Maryland)

**Probabilistic frames**

Dienstag, den 29.05.2012, 16.15 Uhr in MA 313

Abstract:

In this talk I will give an overview of finite frame theory from a probabilistic point of view. In particular, I will review the notion of probabilistic frames and indicate how it is a natural generalization of frames. Concepts such as tight frames, frame potential have their natural counter parts in the setting of probabilistic frames. I will describe these new concepts and will show how probabilistic frames appear in other areas such as statistics, convex geometry, the theory of spherical design, and quantum computing. The talk is based on recent joint work with Martin Ehler.

### Boris Andreianov (U Besançon)

**Existence and finite volume approximation for a cross-diffusion system**

Dienstag, den 5.06.2012, 16.15 Uhr in MA 313

Abstract:

Standard diffusion models in population dynamics may appear inaccurate when compared to the larger family of cross-diffusion models. Yet there are still only few works available for cross-diffusion systems. In this talk, we investigate a class of cross-diffusion system and perform its numerical analysis with the help of a convergent finite volume scheme. The necessary technical tools include the standard arsenal of analysis of nonlinear PDEs (a priori estimates, compactness, identification of nonlinear terms). Numerical analysis requires discrete embeddings and discrete compactness lemmas, that are of interest on their own.

This is a joint work with M. Bendahmane (Bordeaux) and Ricardo Ruiz Baier (Lausanne).

### Zouwei Shen (National U Singapore)

**MRA based wavelet frame and applications**

Dienstag, den 5.06.2012, 17.00 Uhr in MA 313

Abstract:

One of the major driving forces in the area of applied and computational harmonic analysis during the last two decades is the development and the analysis of redundant systems that produce sparse approximations for classes of functions of interest. Such redundant systems include wavelet frames, ridgelets, curvelets and shearlets, to name a few. This talk focuses on tight wavelet frames that are derived from multiresolution analysis and their applications in imaging. The pillar of this theory is the unitary extension principle and its various generalizations, hence we will first give a brief survey on the development of extension principles. The extension principles allow for systematic constructions of wavelet frames that can be tailored to, and effectively used in, various problems in imaging science. We will discuss some of these applications of wavelet frames. The discussion will include frame-based image analysis and restorations, image inpainting, image denoising, image deblurring and blind deblurring, image decomposition, segmentation and CT image reconstruction.

### Hamdullah Yücel (Middle East TU Ankara)

**Adaptive Discontinuous Galerkin Methods for Advection Dominated Optimal Control Problems**

Dienstag, den 12.06.2012, 16.15 Uhr in MA 313

Abstract:

Many real-life applications such as the shape optimization of technological devices, the identification of parameters in environmental processes and flow control problems lead to optimization problems governed by systems of convection diffusion partial differential equations (PDEs). When convection dominates diffusion, the solutions of these PDEs typically exhibit layers on small regions where the solution has large gradients. Hence, it requires special numerical techniques, which take into account the structure of the convection. The integration of discretization and optimization is important for the overall efficiency of the solution process. Discontinuous Galerkin (DG) methods became recently as an alternative to the finite difference, finite volume and continuous finite element methods for solving wave dominated problems like convection diffusion equations since they possess higher accuracy.

This study will focus on analysis and application of DG methods for linear quadratic convection dominated optimal control problems. Because of the inconsistencies of the standard stabilized methods such as streamline upwind Petrov Galerkin (SUPG) on convection diffusion optimal control problems, the discretize-then-optimize and the optimize-then-discretize do not commute. However, the upwind symmetric interior penalty Galerkin (SIPG) method leads to the same discrete optimality systems. The other DG methods such as nonsymmetric interior penalty Galerkin (NIPG) and incomplete interior penalty Galerkin (IIPG) method also yield the same discrete optimality systems when penalization constant is taken large enough. We will study a posteriori error estimates of the upwind SIPG method for the distributed unconstrained and control constrained optimal control problems. In convection dominated optimal control problems with boundary and/or interior layers, the oscillations are propagated downwind and upwind direction in the interior domain, due the opposite sign of convection terms in state and adjoint equations. Hence, we will use residual based a posteriori error estimators to reduce these oscillations around the boundary and/or interior layers. Finally, theoretical analysis will be confirmed by several numerical examples with and without control constraints.

Joint work with Prof. Matthias Heinkenschloss, Rice University and Prof. Bülent Karasözen, Middle East Technical University.

### Daniel Peterseim (HU Berlin)

**Finite Element Computational Homogenization of Multiscale Elliptic Problems**

Dienstag, den 19.06.2012, 16.15 Uhr in MA 313

Abstract:

This talk presents a new approach for computational homogenization of elliptic problems with rough data oscillations in the coefficients without any assumptions on scales. The new (variational) multiscale method is based on a local generalized finite element basis.

Specifically, classical finite element basis functions are corrected by solutions of the variational problem with the additional constraint that some quasi-interpolation of trial and test functions vanishes. This constraint ensures some surprising exponential decay of the corrections and justifies their approximation on local vertex patches. The method represents unresolved scales in such a way that linear convergence with respect to the mesh size is preserved on arbitrary coarse meshes without any pre-asymptotic effects.

Also, everybody is welcome to join the "Nachsitzung" at restaurant / beer garden "Schleusenkrug" at 6 pm.

### Emmanuel Candes (Stanford U)

**Compressive Sensing** (Matheonvortrag)

Mittwoch, den 20.06.2012, 18.00 Uhr in MA 005

Abstract:

Compressive sensing is a new sampling / data acquisition theory based on the discovery that one can exploit sparsity or compressibility when acquiring signals of general interest, and that one can design nonadaptive sampling techniques that condense the information in a compressible signal into a small amount of data. Interestingly, this may be changing the way engineers think about signal acquisition in areas ranging from analog-to-digital conversion, digital optics, magnetic resonance imaging, and seismics. This talk will introduce fundamental conceptual ideas underlying this new sampling or sensing theory. There are already many ongoing efforts to build a new generation of sensing devices based on compressive sensing, and we will address remarkable recent progress in this area as well.

### Chun Liu (Penn State U)

**Energetic Variational Approaches in Complex Fluids**

Dienstag, den 26.06.2012, 16.15 Uhr in MA 313

Abstract:

Complex fluids are ubiquitous in our life. Many complicated but important properties of these materials are due to the couplings and competitions of macroscopic hydrodynamics and underlying microscopic structures. We will introduce some general energetic variational framework in the modeling for these multiscale-multiphysics problems. In particular, the methods allows to distinguish the contributions from the conservative forces and the dissipative forces and derive the constitutive laws which satisfy the laws of thermodynamics.

### Philipp Grohs (ETH Zuerich)

**Algorithmic Treatment of Nonlinear Data**

Dienstag, den 3.07.2012, 16.15 Uhr in MA 313

Abstract:

One of the characteristic features of our modern age is the deluge of data we are confronted with. In addition to the mere masses, we are witnessing more and more modern sensing devices where measurements are of a nonstandard form with data points constrained to nonlinear geometries. With applications ranging from topics in human biomechanics, over image processing, kinematics and robotics to computer graphics, nonlinear geometric data represents a fact of life in modern computational science and therefore it is of eminent interest to develop computational and theoretical tools capable of processing it in an efficient manner. Unfortunately, due to the nonlinear structure inherent in geometric data, classical linear methods known from signal processing, such as wavelet transforms, finite elements, etc., cannot be used in a meaningful way. This calls for genuinely new constructions which respect the underlying geometric structure while satisfying the same desirable properties of well-known linear methods. In my talk I will discuss several such constructions.

### Guy Vallet (U Pau)

**On stochastic conservation laws**

Dienstag, den 3.07.2012, ca 17.00 Uhr in MA 313

Abstract:

In this talk, we are interested in the stochastic perturbation of a multidimensional first-order nonlinear conservation law.

After a brief reminder on conservation laws and stochastic problems, we propose to prove the existence and the uniqueness of the entropy solution.

The result is based on Kruzhkov's doubling-variable method and the convergence in the sense of Young measures.

### Stephan Dahlke (U Marburg)

**The Continuous Shearlet Transform in Arbitrary Space Dimension: General Setting, Shearlet Coorbit Spaces, Traces, and Embeddings**

Dienstag, den 10.07.2012, 16.15 Uhr in MA 313

Abstract:

One of the most important tasks in modern applied mathematics is the analysis of signals. One particular problem which is currently in the center of interest is the extraction of directional information, and several different approaches have been suggested. Among all these approaches, the shearlet transform stands out because it is related to group theory, i.e., it can be derived from a square-integrable representation of a certain group, the full shearlet group.

In the first part of this talk, we are concerned with the generalization of this concept to arbitrary space dimensions. Similar to the two-dimensional case, our approach is based on translations, anisotropic dilations, and specific shear matrices. We show that the associated integral transform again originates from a square-integrable representation of a specific group, the full n-variate shearlet group. Moreover, we verify that by applying the coorbit theory, canonical scales of smoothness spaces, the shearlet coorbit spaces, and associated Banach frames can be derived.

In the second part of the talk, we discuss the structural properties of shearlet coorbit spaces. We show that compactly supported functions with enough smoothness and vanishing moments can serve as analyzing vectors for shearlet coorbit spaces. We use this approach to prove embedding theorems for subspaces of shearlet coorbit spaces resembling shearlets on the cone into Besov spaces. The results are based on general atomic decompositions of Besov spaces. Furthermore, we will discuss trace results for these subspaces with respect to the coordinate planes. It turns out that in many cases these traces are again contained in lower dimensional shearlet coorbit spaces.

This is joint work with G. Steidl (Kaiserslautern) and G. Teschke (Neubrandendburg).

### Martin Gugat (U Erlangen-Nürnberg)

**Boundary Feedback Control of PDE-Systems: Stability Analysis with Lyapunov Functions**

Donnerstag, den 9.08.2012, 16.15 Uhr in MA 313

Abstract:

In many applications in control engineering, the problem to respond to disturbances of the system appears. Typically, the disturbance occurs at some unknown point that is far away from the point where sensing and actuation equipment is installed. It is an interesting challenge to develop good feedback control laws for such cases. We consider systems that can be modeled as 2X2 pdes of hyperbolic type. To show the exponential stability of the system, we work with Lyapunov functions. The Lyapunov functions have to be chosen in such a way that a negative upper bound for their time-derivative that depends linearly on the Lyapunov function can be obtained from the system dynamics.

### Tadele Mengesha (Pennsylvania State U)

**Mathematical Analysis of the linear peridynamic model**

Dienstag, den 21.08.2012, 16.15 Uhr in MA 313

Abstract:

The talk presents a recent work on the mathematical analysis of certain nonlocal models. Our primary example is the peridynamic model of continuum mechanics: a derivative-free, integral-type continuum model that is found to be suitable for modeling materials that naturally form discontinuities such as cracks when deformed. The focus is on the linear bond-based PD model for isotropic elastic materials where we allow an indefinite micromodulus kernel. Our analysis is based on some nonlocal Poincare-type inequalities and compactness of the associated nonlocal operators. We also present the basic structural properties of the associated solution spaces such as compact embedding, separability, completeness and density along with regularity properties of solutions for different types of kernels. Using standard variational techniques we prove the well posedness of the system of equilibrium equations, given as ”nonlocal” boundary value problems. We will also study the Cauchy problem of the time dependent equations of motion with or without damping. Asymptotically, solutions of the nonlocal system are shown to converge to the Navier system of classical elasticity in the event of vanishing nonlocality. (This is a joint work with Qiang Du.)

### Dorothee Knees (WIAS Berlin)

**On a vanishing viscosity approach in damage mechanics**

Dienstag, den 21.08.2012, 17.00 Uhr in MA 313

Abstract:

The aim is to model damage evolution in elastic bodies as a rate-independent process. A well established framework to describe rate-independent processes is the global energetic formulation developed by Mielke and Theil. There, the evolution is characterized via a global stability criterion and an energy balance, which must be satisfied during the whole evolution. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since in many damage models the energy is not simultaneously (strictly) convex in the damage variable and the displacements, one has to deal with solutions being discontinuous in time. The global energetic formulation allows for such solutions. However, due to the global stability criterion, the prediction of the discontinuities often is not satisfactory: global energetic solutions may develop jumps although a local force balance criterion would predict a slow evolution.

The purpose of the lecture is to discuss a vanishing viscosity approach as an alternative for the derivation of a local rate-independent damage model. The starting point is a nonlinear evolution inclusion for the damage variable, which is regularized with a viscosity term. The aim is to study the limit as the viscosity tends to zero. Since the dissipation potential related to damage models is unbounded, one of the challenges is to derive suitable bounds for the thermodynamically conjugated forces (i.e. the derivative of the energy with respect to the damage variable).

This is ongoing joint work with Riccarda Rossi (Brescia) and Chiara Zanini (Torino).

### Inderjit Dhillon (U Texas)

**Orthogonal Matching Pursuit with Replacement**

Mittwoch, den 22.08.2012, 16.15 Uhr in MA 551

Abstract:

In this talk, I will consider the problem of compressed sensing where the goal is to recover sparse vectors using a small number of fixed linear measurements. For this problem, we propose a novel partial hard-thresholding operator that leads to a general family of iterative algorithms. While one extreme of the family yields well known hard thresholding algorithms like ITI (Iterative Thresholding with Inversion) and HTP (Hard Thresholding Pursuit), the other end of the spectrum leads to a novel algorithm that we call Orthogonal Matching Pursuit with Replacement (OMPR). OMPR, like the classic greedy algorithm OMP, adds exactly one coordinate to the support at each iteration, based on the correlation with the current residual. However, unlike OMP, OMPR also removes one coordinate from the support. This simple change allows us to prove that OMPR has the best known guarantees for sparse recovery in terms of the Restricted Isometry Property (a condition on the measurement matrix). Our proof techniques are novel and flexible enough to also permit the tightest known analysis of popular iterative algorithms such as CoSaMP and Subspace Pursuit.

This is joint work with Prateek Jain and Ambuj Tewari.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 464 - everybody's welcome.

### Julian Diaz (U Pau, INRIA Bordeaux)

**Stability analysis of the Interior Penalty Discontinuous Galerkin method for the wave equation**

Dienstag, den 28.08.2012, 16.15 Uhr in MA 313

Abstract:

The Interior Penalty Discontinuous Galerkin Method (IPDGM) is probably one of the the most efficient Discontinuous Galerkin method for solving the wave equation. However, the appropriate determination of the penalization parameter is still an issue, since a too low value leads to unconditionnaly unstable schemes while a too large value severely hampers the CFL condition of the scheme. In the first part of the talk, we show how to determine the penalization parameter in the case of cartesian meshes and we propose an analytical expression of the CFL condition with respect to this parameter. Then, we consider the case of triangular meshes, for which it is not possible to obtain an analytic expression of the penalization parameter. We show, thanks to a numerical study, that it should rather be expressed as a function of the radius of the inscribed circle of the triangles rather than of the radius of the circumscribed circle. Moreover, we propose more accurate expressions based on the angles of the triangles. Finally, we analyze the influence of these different expressions on the CFL condition of the scheme.

### Sebastien Tordeux (U Pau, INRIA Bordeaux)

**Perforated and multiperforated plates in linear acoustic**

Freitag, den 31.08.2012, 9:30 Uhr in MA 415

Abstract:

In a lot of physical problems, the boundary of the computational domain is perforated. This configuration can lead to numerical difficulties when the diameter of the holes are really smaller than the other characteristic lengths. Indeed, it can be very costly to compute a sharp numerical approximation of the solution of such problems for two main reasons: With a standard method like finite elements or finite differences, a refined mesh cannot be avoided in the neighborhood of the hole; the mesh generation of a perforated structure can be a hard task.

Many authors have studied the effect of perforations both from the theoretical and the numerical point of views, see for example [1-4]. In this talk we would like to present some new numerical methods which allows to avoid mesh refinement in the neighborhood of the holes.

References

[1] R. R. Gadyl’shin, Surface potentials and the method of matching asymptotic expansions in the Helmholtz resonator problem, (Russian) Algebra i Analiz 4 (1992), no. 2, 88–115, translation in St. Petersburg Math. J. 4 (1993), no. 2, 273–296.

[2] J. Sanchez-Hubert and E. Snchez-Palencia, Acoustic fluid flow through holes and permeabil- ity of perforated walls, J. Math. Anal. Appl., 87 (1982), pp. 427–453.

[3] A. Taflov, K. Umashanker, B. Becker, F. Harfoush, and K. S. Yee, Detailed fdtd analysis of electromagnetic fields penetrating narrow slots and lapped joints in think conducting screens, IEEE Trans Antenna and Propagation, 36 (1988), pp. 247257.

[4] E. O. Tuck, Matching problems involving flow through small holes, in Advances in applied mechan- ics, Vol. 15, Academic Press, New York, (1975), pp. 89–158.