Research Seminar SoSe 2011

Abstracts

K. Muravleva: Tensor methods for fluid problems

There are many examples of natural and artificial materials behaving as a Bingham medium, i.e., below a certain yield of the stress, the medium behaves as a rigid body, and above this threshold it behaves like an incompressible viscous fluid. Bingham model is a two-parameter model. If in the constitutive relations of a viscoplastic medium it is assumed yield stress limit equal to 0 or viscosity equal to 0, these equations will formally change to the well-known constitutive relations of a viscous fluid or an ideal plastic medium. The main difficulty in the numerical simulation of a viscoplastic medium flow lies in the singularity of the constitutive relations and impossibility to determine the stress in the regions where the strain rate is zero.

Firstly, we compare Newtonian and non-Newtonian fluids, and demonstrate via several test problems (plane Couette flow and the plane, round, and annular Poiseuille flows) differences in behavior of these medium. Later we discuss two main approaches in computations: regularisation and augmented Lagrangian methods (ALM). The regularized viscosity methods consist in the approximation of the constitutive relations by a smooth function that simultaneously describes both the rigid zone and the flow zone. In recent years, the approach to the solution of such problems based on variational inequalities (ALM) has been gaining popularity.

We consider realization of ALM onfinite-difference schemes and apply them to two problems (both steady and unsteady): flow in a pipe (Mosolov's problem) and lid-driven cavity problem.

J. Valdman: Fast MATLAB assembly of FEM matrices in 2D and 3D: nodal elements

We propose an effective and flexible way to assemble finite element matrices in MATLAB. The major loops in the code have been vectorized using the so called array operation in MATLAB, and no low level languages like the C or Fortran has been used for the purpose. The implementation is based on having the vectorization part separated, in other words hidden, from the original code thereby preserving its original structure, and its flexibility as a finite element code. The code is fast and scalable.

http://www.mis.mpg.de/de/publications/andere-reihen/tr/report-1111.html

Lothar Banz: hp-Discontinous Galerkin in Space and Time for Parabolic Obstacle Problems

In this talk we present a hp-FE discontinuous Galerkin method in both space and time for the parabolic obstacle problems. The non-penetration condition is resolved using a Lagrange multiplier yielding a mixed formulation. Its Lagrange multiplier space is spanned by basis functions (in space and time variables) which are biorthogonal to the corresponding basis functions for the primal variable. This biorthogonality allows a component-wise decoupling of the weak contact constraints and can therefore be equivalently rewritten in finding the root of a semi-smooth penalized Fischer-Burmeister non-linear complementary function. The arising system of non-linear equations is solved by a locally Q-quadratic convergent semi-smooth Newton algorithm.

Numerical examples with a 2d space domain underline the strengths and limitations of this approach.

Ulisse Stefanelli: A variational view at linearized plasticity

I shall introduce a variational principle for the quasistatic evolution of a linearized elastoplastic material. The novel characterization allows to recover and partly extend some known time and space-time discretizations convergence results. Moreover, the variational principle is exploited in order to provide some possible a posteriori error control. The talk is based on U. Stefanelli, A variational principle for hardening elastoplasticity, SIAM J. Math. Anal., 40 (2008), 2:623--652.

Jan Valdman (MPI MIS Leipzig): Functional a posteriori error estimates of elastic problems with nonlinear boundary conditions

The talk is based on Pekka Neittaanmäki, Sergey Repin, Jan Valdman: "Functional a posteriori error estimates for elastic problems with nonlinear boundary conditions", Preprint 13/2011 MPI MIS Leipzig

Ilaria Perugia: Trefftz-Discontinuous Galerkin Methods for Wave Problems

Several finite element methods used in the numerical discretization of wave problems in frequency domain are based on incorporating a priori knowledge about the differential equation into the local approximation spaces by using Trefftz-type basis functions, namely functions which belong to the kernel of the considered differential operator. For the Helmholtz equation, for instance, examples of Trefftz basis functions are plane waves, Fourier-Bessel functions and Hankel functions, and there are in the literature several methods based on them: the Plane Wave/Bessel Partition of Unit Method by Babuška and Melenk, the Ultra Weak Variational Formulation by Cessenat and Després, the Plane Wave/Bessel Least Square Method by Monk and Wang, the Discontinuous Enrichment Method by Farhat and co-workers, the Method of Fundamental Solutions by Stojek, to give some examples. These methods differ form one another not only for the type of Trefftz basis functions used in the approximating spaces, but also for the way of imposing continuity at the interelement boundaries: partition of unit, least squares, Lagrange multipliers or discontinuous Galerkin techniques.

In this talk, the construction of Trefftz-discontinuous Galerkin methods for both the Helmholtz and the time-harmonic Maxwell problems will be reviewed and their abstract error analysis will be presented. It will also be shown how to derive best approximation error estimates for Trefftz functions, needed to complete the convergence analysis, by using Vekua's theory. Some explicit estimates in the case of plane waves will be given. These results have been obtained in collaboration with Ralf Hiptmair and Andrea Moiola form ETH Zürich.