TU Dortmund




Adresse (Briefe):

Technische Universität Dortmund
Fak. Mathematik, LS X
44221 Dortmund

Adresse (Lieferungen):

Technische Universität Dortmund
Fak. Mathematik, LS X
Vogelpothsweg 87
44227 Dortmund

Telefonnummern und Email-Adressen:

Fr. C. Mecke (Sekretariat):
(+49) 231 / 755-5411

Prof. Dr. H. Blum:
(+49) 231 / 755-5410

Prof. Dr. Ch. Kreuzer:
(+49) 231 / 755-5425

Prof. Dr. Ch. Meyer:
(+49) 231 / 755-5412

Fax: (+49) 231 / 755-5416


Aktuelle Informationen

Vortrag am 19.04.2018 im Rahmen des Oberseminars Numerische Analysis und Optimierung

Vortragender: Dr. Pietro Zanotti, TU Dortmund

Zeit: 14.00 Uhr c.t.

Ort: Mathematikgebäude, Raum M 614

Titel: Quasi-optimal nonconforming methods for symmetric elliptic problems

Abstract: In this lecture, I will consider the approximation of linear variational problems, symmetric and elliptic for simplicity. According to the classical Céa's lemma, the approximation of these problems by a conforming Ritz-Galerkin method is quasi-optimal, in the sense that the error of the method, measured in the energy norm, equals the best approximation error. In contrast, a simple argument reveals that many classical nonconforming methods (like the Crouzeix-Raviart or Discontinuou Galerkin methods) do not enjoy such property. Motivated by this observation, I will introduce a rather large class of possibly nonconforming methods and I will characterize the subclass of the quasi-optimal ones. For this purpose, I will propose notions of stability and consistency that are necessary and sufficient for quasi-optimality. The size of the constants involved in the analysis will also be discussed. To illustrate the abstract results, I will describe a modified version of the Crouzeix-Raviart method that is quasi-optimal for the Poisson problem.

The lecture will cover a selection of arguments from my PhD thesis, which was written at the Università degli Studi di Milano, under the supervision of Prof. Andreas Veeser.

Vortrag am 26.04.2018 im Rahmen des Oberseminars Numerische Analysis und Optimierung

Vortragender: Prof. Georgoulis, Universität Leicester

Zeit: 14 Uhr c.t.

Ort: Mathematikgebäude, Raum M 614

Title: Recent developments on finite element methods on meshes consisting of polygonal and/or polyhedral elements

Abstract: Numerical methods defined on computational meshes consisting of polygonal and/or polyhedral (henceforth collectively termed as “polytopic") elements, with, potentially, many faces, have gained substantial traction in recent years for a number of important reasons. Clearly, a key underlying issue for all classes of FEM is the design of a suitable computational mesh upon which the underlying PDE problem will be discretized. The task of generating the mesh must address two competing issues: on the one hand, the mesh should provide a good representation of the given computational geometry with suffi- cient resolution for the computation of accurate numerical approximations. On the other hand, the mesh should not be so fine that computational turn-around times are too high, or in some cases even intractable, due to the high number of degrees of freedom in the resulting FEM. Traditionally, standard mesh generators generate grids consisting of triangular/quadrilateral elements in 2D and tetrahedral/hexahedral/prismatic/pyramidal elements in 3D. In the presence of essentially lower-dimensional solution features, for example, boundary/internal layers, anisotropic meshing may be exploited. However, in regions of high curvature, the use of such highly-stretched elements may lead to element self-intersection, unless the curvature of the geometry is carefully ‘propagated’ into the interior of the mesh through the use of (computationally expensive) isoparametric element mappings. These issues are particularly pertinent in the context of high-order methods, since in this setting, accuracy is often achieved by exploiting coarse meshes in combination with local high-order polynomial basis functions. I will argue that, by dramatically increasing the flexibility in terms of the set of admissible element shapes present in the computational mesh, the resulting, possibly discontinuous, FEMs can potentially deliver dramatic savings in computational costs. Moreover, I will present some recent theoretical developments in the error analysis of both “conforming” and discontinuous Galerkin finite element methods.