Seminari Dipartimento di Matematica - Politecnico di Milano

28 Giugno, 2010 14:30 in punto

A new class of discontinuous PETROV-GALERKIN (DPG) finite element (FE) methods with application to convection-dominated diffusion problems

Leszek Demkowicz, The University of Texas at Austin

Aula Seminari F. Saleri VI Piano MOX- Dipartimento di Matematica, Politecnico di Milano

The hp-adaptive finite elements combine elements of varying size h and polynomial order p to deliver approximation properties superior
to any other discretization methods. The best approximation error converges exponentially fast to zero as a function of number of degrees-of-freedom. The hp methods are thus a natural candidate for singularly perturbed problems experiencing internal or boundary layers like in compressible gas dynamics.

This is the good news. The bad news is that only a small number of variational formulations is stable for hp-discretizations.
By the hp-stability we mean a situation where the discretization error can be bounded by the best approximation error times a constant that is independent of both h and p. To this class
belong classical elliptic problems (linear and non-linear), and a large class of wave propagation problems whose discretization
is based on hp spaces reproducing the classical exact grad-curl-divsequence. Examples include acoustics, Maxwell, elastodynamics,poroelasticity and various coupled and multiphysics problems.

We will present a new paradigm for constructing discretization schemes for virtually arbitrary systems of linear PDE s that remain stable for arbitrary hp meshes, extending thus dramatically
the applicability of hp approximations. For a start, we focus on a challenging model problem - convection dominated diffusion.

The presented methodology incorporates the following features:

The problem of interest is formulated as a system of first order PDE s in the distributional (weak) form, i.e. all derivatives are moved to test functions. We use the DG setting, i.e. the
integration by parts is done over individual elements.

As a consequence, the unknowns include not only field variables within elements but also fluxes on interelement boundaries. We do not use the concept of a numerical flux but, instead, treat the fluxes as independent, additional unknowns.

For each trial function corresponding to either field or flux variable, we determine a corresponding optimal test function by solving an auxiliary local problem on one element.
The use of optimal test functions guarantees attaining the supremum in the famous inf-sup condition from Babuska-Brezzi theory.

The local problems for determining optimal test functions are solved approximately with an enhanced approximation (a locally
hp-refined mesh).

By selecting right norms for test functions, we can obtain amazing stability properties uniform not only with respect to discretization
parameters but also with respect to the diffusion constant (the resulting discretization is robust).

The presentation will consist of three parts.

In the first part, we will present a general abstract variational
framework for the optimal test functions.

The second part will focus on the analysis of 1D convection-dominated diffusion
problem. We will present a 1D stability analysis.

The third part will focus on adaptivity. We will present numerous numerical examples for 1D and 2D confusion problems. We have been able to solve in a fully automatic mode problems with diffusion constant eps = 10^{-11} in 1D
and eps = 10^{-7} in 2D using hp-adaptivity.

The presentations will be accompanied with live demonstrations of 1D and 2D codes.

22 Giugno, 2010 14:30 in punto

Sezione di Analisi

L^p and Schauder estimates for nonvariational operators structured on Hörmander’s vector fields with drift

Maochun Zhu, Northwestern Polytechnical University, Xi an, Shaanxi, CHINA

Dipartimento di Matematica del Politecnico di Milano, Via Bonardi 9, aula seminari 3° piano

17 Giugno, 2010 14:30 in punto

Sezione di Analisi

Electrode Reconstruction by FIB/SEM and Microstructure Modelling - Optimal experimental design for PDE

Thomas Carraro, Universita di Heidelber

Aula Seminari F. Saleri VI Piano MOX- Dipartimento di Matematica, Politecnico di Milano

abstract

The talk consists of two distinct parts, presenting the main topics of my research interests. In the first part I present a FEM model for the detailed simulation of an
electrode microstructure used in the most performing SOFC (Solid Oxide Fuel Cells). This is a joint work together with the IWE group of Prof. Ivers-Tiff´ee at KIT (Karlsruhe Institute of Technology).

Brief description: As polarization losses within SOFC electrodes are strongly related to the material microstructure and composition, a high-resolution microstructure analysis is a sensible method to understand and improve electrode performance.
By means of a dual-beam focused ion beam/scanning electron microscope (FIB/SEM) we obtain a 3D reconstruction of a high performance LSCF-cathode. These data are the basis for the calculation of the microstructure parameters like (i) surface area, (ii) volume/porosity fraction and (iii) tortuosity.
With a detailed FEM model we can calculate the electrode performance (Area Specific Resistance ASR) from the 3D reconstruction.

In the second part of the talk I give a short introduction on optimal experimental design (OED) in the context of PDE.

Brief description: The goal of the optimal experimental design (OED) is a robust prediction of the model parameters by an appropriate choice of the design (setup) of the experiments. In the OED context a robust prediction is an estimation of the parameters with the smallest variance.
In the framework of PDE this is a complex optimization task for different reasons.
On one side, it corresponds in general to a PDE-constrained minimization of a nonlinear functional, which depends on the derivatives of the state of the simulated system. On the other side, the best performing and robust techniques
are needed to solve numerically the huge problem deriving from the discretization of the system of equations. In this talk I consider the case of finite dimensional parameter space and present a numerical approach to the optimization of the
design based on a finite element method for the solution of the state equation and the sensitivity equations needed to calculate the covariance matrix of the parameters.