## Doktorandenseminar des WIAS

# Numerische Mathematik und Wissenschaftliches Rechnen

Forschungsgruppe

zurück zum aktuellen Programm

## Seminar Numerische Mathematik / Numerical mathematics seminars

aktuelles Programm / current program

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Donnerstag, 03. 12. 2015, 14:00 Uhr (WIAS-406)
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Dr. A. Linke(WIAS Berlin)

Towards pressure-robust mixed methods for the incompressible Navier--Stokes equationsMixed methods for the incompressible Navier-Stokes equations are reviewed with respect to the discretization of the divergence constraint. Though the establishment of inf-sup stable mixed methods represents a milestone in the development of discretization theory for flow problems, many important questions are left open, and classical text books usually convey a wrong impression what are they best qualitatively possible results, which are achievable in the field. Especially, it will be shown that the construction of pressure-robust mixed methods, whose velocity error is pressure-independent, is rather easy, though this was thought to be nearly impossible for many years. Numerical examples will show that classical mixed methods deliver poor results, whenever large irrotational forces appear in the Navier-Stokes momentum balance, while pressure-robust mixed methods perform well.

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Donnerstag, 26. 11. 2015, 14:00 Uhr (WIAS-ESH)
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Dr. F. Dassi(WIAS Berlin)

Achievements and challenges in anisotropic mesh generation

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Donnerstag, 19. 11. 2015, 10:00 Uhr (ESH)
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Prof. Ch. Pflaum(Friedrich-Alexander Universität Erlangen-Nürnberg)

Discretization of elliptic differential equations with variable coefficients on sparse gridsSparse grids are discretization grids which can be used to reduce the computational amount for solving partial differential equations. Using the Galerkin discretization, one obtains a linear equation system with O(N (log N)d-1) unknowns. The corresponding discretization error is O(N-1 (log N)d-1) in the H1-norm. A major difficulty in using this sparse grid discretization is the complexity of the related stiffness matrix. As a consequence only PDE?s with constant coefficients can be efficiently be discretized using the standard sparse grid discretization with d>2. To reduce the complexity of the sparse grid discretization matrix, we apply prewavelets. This simplifies the implementation of the corresponding algorithms. Furthermore, we present a new sparse grid discretization for the discretization of elliptic differential equations with variable coefficients. This discretization utilizes a semi-othogonality property. The convergence rate and stability of the discretization is proven for arbitrary dimensions d.

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Donnerstag, 05. 11. 2015, 14:00 Uhr (ESH)
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J. Pellerin(WIAS Berlin)

Simultaneous meshing and simplification of complex 3D geometrical models using Voronoi diagramsAll meshing methods aim at dividing a physical model that has a potentially very complex geometry into simple elements. Requirements on the resulting mesh might be contradictory as the elements shall conform to the shape of the physical domain and shall satisfy constraints on their shapes, sizes, number etc. In geological modeling to generate robustly volumetric meshes, it is often necessary to adapt the level of detail of the geological domain when meshing their boundary surfaces. In the first part of the talk, I will focus on geological structural models, their geometrical and geological specificities, and the induced challenges for meshing methods. Then, I will detail the key ideas of a meshing method that tackles these challenges:

(1) the use of a well-shaped Voronoi diagram to subdivide the model and

(2) the combinatorial considerations to build well-shaped mesh elements from the connected components of the intersections of the Voronoi cells/facets/edges/vertices with the model entities.

This approach allows modifications of the input model, a crucial point for geomodeling applications. Finally, I will discuss shortly the implementation of the method.

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Donnerstag, 24. 09. 2015, 10:00 Uhr (ESH)
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Yueyuan Gao(Universite Paris-Sud, France)

Finite volume methods for first order stochastic conservation lawsWe perform Monte-Carlo simulations in the one-dimensional torus for the first order Burgers equation forced by a stochastic source term with zero spatial integral. We suppose that this source term is a white noise in time, and consider various regularities in space. We apply a finite volume scheme combining the Godunov numerical flux with the Euler-Maruyama integrator in time. It turns out that the empirical mean converges to the space-average of the deterministic initial condition as $t\rightarrow\infty$. The empirical variance also stabilizes for large time, towards a limit which depends on the space regularity and on the intensity of the noise.

We then study a time explicit finite volume method with an upwind scheme for a conservation law driven by a multiplicative source term involving a $Q$-Wiener process. We present some a priori estimates including a weak BV estimate. After performing a time interpolation, we prove two entropy inequalities for the discrete solution and show that it converges up to a subsequence to a stochastic measure-valued entropy solution of the conservation law in the sense of Young measures.

The numerical part is joint work with E. Audusse and S. Boyaval while the convergence proof is joint work with T. Funaki and H. Weber.

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Freitag, 11. 9. 2015, 14:00 Uhr (ESH)
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Dr. T. Benacchio(Met Office, UK)

Towards scalable numerical weather and climate prediction with mixed finite element discretizationsThe nature of atmospheric processes presents a unique set of challenges to numerical modelling, as coupled nonlinear phenomena evolve on a wide range of spatial and temporal scales extending across several orders of magnitude.. Timely and reliable forecasts require accurate simulation of transport processes together with robust handling of waves and balanced regimes, with competitive and scalable performance a key requisite on increasingly parallel architectures. Features of the nonhydrostatic compressible dynamical core in operation at the Met Office will be outlined, with particular reference to the scalability bottleneck given by the currently employed latitude-longitude grid structure. The talk will then focus on theoretical and computational aspects and open challenges of the mixed finite element numerical scheme on quasi-uniform spherical grids underpinning the next generation dynamical core.

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Donnerstag, 03. 09. 2015, 14:00 Uhr (ESH)
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Dr. G. R. Barrenechea(University of Strathclyde, UK)

Stabilising some inf-sup stable pairs on anisotropic quadrilateral meshesThe finite element solution of the Stokes problem is subject to the well-known inf-sup condition. This condition usually depends on the domain and the degree of the polynomial spaces used in the discretisation. Now, when anisotropic finite elements are used, this inf-sup condition usually degenerates with the aspect ratio. In this talk I will present results identifying the part of the pressure space that is responsible for this degeneration, and present a way to solve that. In the second part of the talk, this technique will be applied to some, non inf-sup stable this time, pairs of spaces for the Stokes and Oseen equations. The work presented in this talk is in collaboration with Mark Ainsworth (Brown) and Andreas Wachtel (Strathclyde).

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Dienstag, 01. 09. 2015, 13:30 Uhr (ESH)
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Dr. P. Knobloch(Charles University, Czech Republic)

On linearity preservation and discrete maximum principle for algebraic flux correction schemesWe consider a general algebraic flux correction scheme for linear boundary value problems in any space dimension and prove its solvability. The properties of the scheme depend on the definition of limiters used to limit fluxes that would otherwise cause spurious oscillations. We give an example of limiters that are often used in computations. For these limiters we prove the discrete maximum principle but we also show that they generally do not lead to a linearity preserving method. Therefore, we propose another limiters for which the linearity preservation is satisfied and the discrete maximum principle holds for arbitrary meshes.

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Donnerstag, 25. 6. 2015, 14:00 Uhr (ESH)
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S. Hirsch(Charite - Universitätsmedizin Berlin)

Compression-sensitive Magnetic Resonance Elastography and poroelasticityRecent research suggests that the compression modulus of organic tissue is sensitive to changes of tissue pressure. Compression-sensitive Magnetic Resonance Elastography (MRE) provides a means to map the amplitudes of externally stimulated mechanical pressure waves and has proven its potential to differentiate between physiological states of different tissue pressure. The interpretation of the results thus far has relied on an effective medium model. Biot's poroelastic theory can provide an explanation for the interaction between tissue pressure and the mechanical properties of tissue. However, quantification of the poroelastic parameters from MRE data still poses a challenge. This talk will present the status quo of the technique and explore the potential for future improvements towards the ultimate goal of non-invasive pressure measurement.

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Dienstag, 16. 6. 2015, 13:30 Uhr (ESH)
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Prof. S. Ganesan(Indian Institute of Science)

Finite element algorithms for massively parallel architecturesParallel algorithms with hybrid MPI and OpenMP implementations for a geometric multigrid solver in a nite element scheme will be presented in this talk. In particular, the mesh partitioning, the nite element mappers and communicators on a hierarchy of multigrid mesh will be discussed. Further, a multicolor strategy in the smoothing steps of the multigrid solver, parallel restrictions and prolongations using halo cells will be discussed.

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Donnerstag, 11. 6. 2015, 14:00 Uhr (ESH)
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Prof. J. Novo(Universidad Autonoma de Madrid, Spain)

Local error estimates for the SUPG method applied to evolutionary convection-reaction-diffusion equationsLocal error estimates for the SUPG method applied to evolutionary convection-reaction-diffusion equations are considered. The steady case is reviewed and local error bounds are obtained for general order finite element methods. For the evolutionary problem, local bounds are obtained when the SUPG method is combined with the backward Euler scheme. The arguments used in the proof lead to estimates for the stabilization parameter that depend on the length on the time step. The numerical experiments show that local bounds seem to hold true both with a stabilization parameter depending only on the spatial mesh grid and with other time integrators.

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Donnerstag, 4. 6. 2015, 14:00 Uhr (ESH)
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Dr. K. Schmidt(TU Berlin)

On optimal basis functions for thin conducting sheets in electromagnetics and on efficient calculation of the photonic crystal bandstructureThe talk consists of two parts.

The first part of the talk is dedicated to thin conducting sheets which are commonly used in the protection of electronic devices. The domain of computation is the thin metallic sheet, the surrounding air and other metallic or isolating parts. With their large aspect ratios the shielding sheets become a serious issue for the direct application of the finite element method (FEM) on triangular or tetrahedal cells. In addition the sheets exhibit boundary layers (this is called skin effect) whose size depends on the frequency and may become smaller than the sheet thickness. We propose a semi-discretisation inside the sheet as a tensor product of $H^1(\Gamma)$-bounded functions on the midline $\Gamma$ and a family of $N$ basis functions in thickness direction. The basis functions are defined hierarchically in the spirit of Vogelius and Babuska by an hierarchy of ordinary differential equations in thickness directions. For each basis function a new term on $\Gamma$ enters the variational formulation. We give explicit formulas for the basis functions for straight and curved sheets, prove the well-posedness and estimate for straight sheets the modelling error. If the sheet surfaces and so the trace of the exact solution is analytic then this semi-discretisation leads to an exponential convergence in $N$, whose rate is independent of the frequency. This will be illustrated by numerical experiments for straight and circular sheet.

In the second part the calculation of the photonic crystal bandstructure is studied. Photonic crystals are periodic dielectric material, a structure which has high influence on the propagation of light. We are interested on the related parametric eigenvalue problem where the dispersion curves are the functions of the eigenvalues in dependence of the parameter. Based on the Fredholm theory we explicit formulas for the first and higher derivatives of the dispersion curves in terms of eigenvalue and eigenfunction. The derivatives will be used for an adaptive parameter sampling and high-order description of the band structure based on Taylor expansions. The algorithm is able to decide whether two bands cross or not.

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Donnerstag, 28. 5. 2015, 14:00 Uhr (ESH)
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Dr. P. A. Zegeling(Utrecht University, The Netherlands)

Adaptive grids for detecting non-monotone waves and instabilities in a non-equilibrium PDE model from porous mediaSpace-time evolution described by nonlinear PDE models involves patterns and qualitative changes induced by parameters. In this talk I will emphasize the importance of both the analysis and computation in relation to a bifurcation problem in a non-equilibrium Richard's equation from hydrology. The extension of this PDE model for the water saturation $S$ to take into account additional dynamic memory effects was suggested by Hassanizadeh and Gray in the 90's. This gives rise to an extra \emph{third-order mixed} space-time derivative term in the PDE of the form $\tau ~ \nabla \cdot [T(S) \nabla (S_t)]$.

In one space dimension traveling wave analysis is able to predict the formation of steep non-monotone waves depending on $\tau$. In 2D, the parameters $\tau$ and the frequency $\omega$ included in a small perturbation term, predict that the waves may become \emph{unstable}, thereby initiating so-called gravity-driven fingering structures. This phenomenon can be analysed with a linear stability analysis and its effects are supported by the numerical experiments of the 2D time-dependent PDE model. For this purpose, we have used a sophisticated adaptive grid r-refinement technique based on a recently developed monitor function. The numerical experiments in one and two space dimension show the effectiveness of the adaptive grid solver.

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Donnerstag, 30. 4. 2015, 14:00 Uhr (ESH)
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Dr. A. Caiazzo(WIAS Berlin)

Stabilization at backflowIn computational fluid dynamics, the presence of incoming flow at open boundaries (backflow) often yields to unphysical instabilities. This issue arises due to the incoming convective energy at the open boundary, which - in general cases - cannot be controlled a priori. In this talk, the state-of-the-art of backflow stabilizations will be overviewed, discussing then two recently proposed approaches. The first is based on adding artificial viscosity on the open boundary along the tangential direction, while the second consists in penalizing the weak residual of a Stokes problem on the boundary. The performance of the methods are assessed through several numerical tests, considering analytic solutions, as well as blood and air flows in complex geometries coming from medical images.

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Donnerstag, 16. 4. 2015, 14:00 Uhr (ESH)
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L. Heltai(SISSA, Italy)

Coupling isogeometric analysis and Reduced Basis methods for complex geometrical parametrizationsIsogeometric analysis (IGA) emerged as a technology bridging Computer Aided Geometric Design (CAGD), most commonly based on Non-Uniform Rational B-Splines (NURBS) surfaces, and engineering analysis.

In finite element and boundary element isogeometric methods (FE-IGA and IGA-BEM), the NURBS basis functions that describe the geometry define also the approximation spaces. The resulting approximation schemes can be used very effectively as a high-fidelity approximation in Reduced Basis (RB) methods, providing a tool for the rapid and reliable evaluation of PDE systems characterized by complex geometrical features.

After a brief overview of the various techniques, I will present an application of this technology, where we address the simulation of potential flows past airfoils, parametrized with respect to the angle of attack and the NACA number identifying their shape, to optimize in real time the trimming of the rigid sail of a catamaran, using RB IGA-BEM.

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Dienstag, 3. 3. 2015, 13:30 Uhr (ESH)
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Prof. S. Perotto(Politecnico di Milano, Italy)

Adaptive Hierarchical Model (HiMod) reduction for initial boundary value problemsThe construction of surrogate models is a crucial step for bringing computational tools to practical applications within an appropriate timeline. This can be accomplished by taking advantage of specific features of the problem at hand. For instance, when solving flow problems in networks (in the modeling of blood, oil, water, or air dynamics), the local dynamics are expected to develop mainly along a dominant direction. The interaction between the local and network dynamics calls often for appropriate model reduction techniques. This is the case, for instance, of the so-called Hierarchical Model (HiMod) reduction, devised to deal with problems characterized by a prevalent dynamics, even though the presence of transverse dynamics may be locally significant. The main idea of HiMod consists of introducing a modal discretization for the transverse dynamics coupled with a finite element approximation of the mainstream one [1]. Moving from the original approach where we employ the same number of modes on the whole domain, we have successively introduced a more sophisticated approach, by automatically adapting the number of the modal functions according to the local level of complexity of the problem. This goal is pursued by deriving an a posteriori modelling error analysis, first set in a steady framework [2] and more recently extended to an unsteady setting [3]. An adaptive selection of the space and of the space-time mesh is also accomplished.

Reference:

[1] S. Perotto, A. Ern and A. Veneziani, ''Hierarchical local model reduction for elliptic problems: a domain decomposition approach''. Multiscale Model. Simul., 8 (2010), no. 4, 1102-1127.

[2] S. Perotto and A. Veneziani, ''Coupled model and grid adaptivity in hierarchical reduction of elliptic problems''. J. Sci. Comput., 60 (2014), no. 3, 505-536.

[3] S. Perotto and A. Zilio, ''Space-time adaptive hierarchical model reduction for parabolic equations''. MOX Report no. 06/2015.

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Dienstag, 10. 2. 2015, 13:30 Uhr (ESH)
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Prof. R. Masson(Universite de Nice Sophia Antipolis, France)

Gradient scheme discretizations of two phase porous media flows in fractured porous mediaThis talk presents the gradient scheme framework for the discretization of two-phase Darcy flows in discrete fracture networks taking into account the mass exchange between the matrix and the fracture. We consider the asymptotic model for which the fractures are represented as interfaces of codimension one immersed in the matrix domain, leading to the so called hybrid dimensional Darcy flow model. The pressures at the interfaces between the matrix and the fracture network are continuous corresponding to a ratio between the normal permeability of the fracture and the width of the fracture assumed to be large compared with the ratio between the permeability of the matrix and the size of the domain. Two type of discretizations matching the gradient scheme framework are discussed including the extensions of the Hybrid Finite Volume (HFV) and of the Vertex Approximate Gradient (VAG) schemes to the case of hybrid dimensional Darcy flow models. Compared with Control Volume Finite Element (CVFE) approaches, the VAG scheme has the advantage to avoid the mixing of the fracture and matrix rock types at the interfaces between the matrix and the fractures, while keeping the low cost of a nodal discretization on unstructured meshes. The convergence of the gradient schemes is obtained for two phase flow models under the assumption that the relative permeabilities are bounded from below by a strictly positive constant. This assumption is needed in the convergence proof only in order to take into account discontinuous capillary pressures in particular at the matrix fracture interfaces. The efficiency of our approach is shown on numerical examples of fracture networks in 2D and 3D.

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Donnerstag, 22. 1. 2015, 14:00 Uhr (ESH)
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S. Rubino(Universidad de Sevilla, Spain)

Finite element approximation of an unsteady projection-based VMS turbulence model with wall lawsIn this work we present the numerical analysis and study the performance of a finite element projection-based Variational MultiScale (VMS) turbulence model that includes general non-linear wall laws. We introduce Lagrange finite element spaces adapted to approximate the slip condition. The sub-grid effects are modeled by an eddy diffusion term that acts only on a range of small resolved scales. Moreover, high-order stabilization terms are considered, with the double aim to guarantee stability for coarse meshes, and help to counter-balance the accumulation of sub-grid energy together with the sub-grid eddy viscosity term. We prove stability and convergence for solutions that only need to bear the natural minimal regularity, in unsteady regime. We also study the asymptotic energy balance of the system. We finally include some numerical tests to assess the performance of the model described in this work.