Seminár z kvalitatívnej teórie diferenciálnych rovníc - Hana Mizerová (17.12.2015)

vo štvrtok 17.12.2015 o 14:00 hod. v miestnosti M/223

19. 11. 2015 09.42 hod.
Od: Pavol Quittner

Prednášajúci: Hana Mizerová (Johannes Gutenberg University in Mainz)

Názov prednášky: Analysis and numerical solution of the Peterlin viscoelastic model

Termín: 17.12.2015, 14:00 hod., M/223

We introduce a mathematical model describing the unsteady motion of an incompressible polymeric fluid. Each polymer molecule is treated as two beads connected by a spring. For the nonlinear spring force it is not possible to obtain a closed system of equations for the conformation tensor, unless we approximate the force law. The Peterlin approximation replaces the length of the spring by the length of the average spring, cf. [1]. Consequently, the macroscopic dumbbell-based model for dilute polymer solutions is obtained. The model consists of the conservation of mass and momentum and time evolution of the symmetric positive definite conformation tensor, where the diffusive effects are taken into account. Using the energy estimates we prove, in two space dimensions, global in time existence of weak solutions. Assuming more regular data we show higher regularity and consequently uniqueness of the weak solution, cf. [2]. For the Oseen-type Peterlin model we propose a linear pressure-stabilized characteristics finite element scheme.
We derive the corresponding error estimates and we prove, for linear finite elements, the optimal first order accuracy. Theoretical error of the numerical approximation is confirmed by a series of numerical experiments, see [3].

The present work has been supported by the German Science Foundation (DFG) under IRTG 1529  "Mathematical Fluid Dynamics" and TRR 146 "Multiscale Simulation Methods for Soft Matter Systems". 
It has been realized in collaboration with M. Lukáčová, Š. Nečasová, M. Renardy and H. Notsu, M. Tabata.

[1] M. Renardy, Mathematical analysis of viscoelastic flows, CBMS-NSF Conference Series in Applied Mathematics 73, SIAM (2000).
[2] M. Lukáčová, H. Mizerová and Š. Nečasová, Global existence and uniqueness result for the diffusive Peterlin viscoelastic model, Nonlinear Anal. 120 (2015), 154-170.
[3] M. Lukáčová, H. Mizerová, H. Notsu and M. Tabata, Error estimates of a pressure-stabilized characteristics finite element method for the Oseen-type Peterlin model, in preparation.