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Analysis of finite element methods for incompressible flow and for elasticity equations

Lecturer

Associate Professor Ricardo Ruiz Baier, Monash University

Synopsis

This course will develop, analyse, and apply the mathematical theory of finite element discretisations for the equations of incompressible flow and elasticity. Prominent examples are the Navier-Stokes equations and the equations of linear elasticity. We will review the analysis of weak solutions to Stokes and Navier-Stokes equations using the abstract theory of saddle-point problems. We will introduce discretisations based on finite element methods, and analyse the stability of the resulting numerical schemes. Extensions to other formulations (using stress and vorticity) will be addressed. The contents also cover computational algorithms and hands-on implementation of increasingly complex problems using modern open-source libraries. It is intended primarily for honours and other graduate students interested in the numerical solution of problems in continuum mechanics and their rigorous mathematical properties.

Course Overview

Topics covered in this course include:

  • Introduction: Balance laws. Stokes, elasticity, and Navier-Stokes equations. Discussion on simplified models and on numerous physical examples which underline such systems of equations. Classification of equations.
  • Solvability analysis: Weak formulations. Discussion on why the basic theory for elliptic problems is insufficient. Mixed formulations and non-coercive partial differential equations. The inf-sup condition and its application to the Stokes and Darcy problems.
  • Mixed finite element methods: Galerkin approximation. Stability. Relationship with the continuous problem. Discrete inf-sup conditions also in the algebraic setting. Convergence properties, a priori error estimates in different norms. Solution of linear algebraic systems. Efficient implementation.
  • Increasing complexity: Incorporation of non-linearities. Designing formulations using secondary unknowns (vorticity, rotations, stress, strain).

The course material  is complemented by a balanced set of theoretical and practical problems, working also on the algorithmic and computer implementation aspects and using easy-to-use scripts in python/Julia.

Prerequisites

  • Essential: Analysis, Numerical methods, Partial differential equations
  • Desirable: Continuum mechanics, Functional analysis, Programming skills in Python/Julia

Assessment

A short quiz covering theoretical concepts of a mixed formulation for incompressible flow and its numerical implementation.

(Final assessment details to be confirmed)

Attendance requirements

  • For those completing the subject for their own knowledge/interest, final attendance requirements will be confirmed

Resources/pre-reading

Banach and Hilbert spaces, Weak solutions of elliptic PDEs, Basic concepts in numerical analysis (stability, interpolation, convergence).

Some references:

  • A. Quarteroni. Numerical Models for Differential Problems. Springer 2009;
  • A. Ern, J-L. Guermond, Theory and Practice of Finite Elements. Springer 2004.

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Associate Professor Ricardo Ruiz Baier, Monash University

Ricardo works on the design and analysis of numerical methods for partial differential equations. His areas of interest and expertise also include fundamental topics in scientific computing, as well as a number of applications such as multiphase flow and transport in porous media and cardiac electromechanics. Ricardo is an Associate Professor in Computational Mathematics at Monash Uni. Before joining Monash, Ricardo spent four years as Lecturer in Numerical Analysis at the Mathematical Institute of the University of Oxford. Previously, he was a Senior Researcher at the University of Lausanne, and a Postdoctoral Fellow at the Ecole Polytechnique Federale de Lausanne. He is originally from Chile.