Finite element methods form a flexible class of techniques for numerical solution of PDEs that are both accurate and efficient.
The finite element method is a core mathematical technique underpinning much of the development of simulation science. Applications are as diverse as the structural mechanics of buildings, the weather forecast, and pricing financial instruments. Finite element methods have a powerful mathematical abstraction based on the language of function spaces, inner products, norms and operators.
This module aims to develop a deep understanding of the finite element method by spanning both its analysis and implementation. in the analysis part of the module you will employ the mathematical abstractions of the finite element method to analyse the existence, stability, and accuracy of numerical solutions to PDEs. At the same time, in the implementation part of the module you will combine these abstractions with modern software engineering tools to create and understand a computer implementation of the finite element method.
• Basic concepts: Weak formulation of boundary value problems, Ritz-Galerkin approximation, error estimates, piecewise polynomial spaces, local estimates.
• Efficient construction of finite element spaces in one dimension, 1D quadrature, global assembly of mass matrix and Laplace matrix.
• Construction of a finite element space: Ciarlet’s finite element, various element types, finite element interpolants.
• Construction of local bases for finite elements, efficient local assembly.
• Sobolev Spaces: generalised derivatives, Sobolev norms and spaces, Sobolev’s inequality.
• Numerical quadrature on simplices. Employing the pullback to integrate on a reference element.
• Variational formulation of elliptic boundary value problems: Riesz representation theorem, symmetric and nonsymmetric variational problems, Lax-Milgram theorem, finite element approximation estimates.
• Computational meshes: meshes as graphs of topological entities. Discrete function spaces on meshes, local and global numbering.
• Global assembly for Poisson equation, implementation of boundary conditions. General approach for nonlinear elliptic PDEs.
• Variational problems: Poisson’s equation, variational approximation of Poisson’s equation, elliptic regularity estimates, general second-order elliptic operators and their variational approximation.
• Residual form, the Gâteaux derivative and techniques for nonlinear problems.
The course is assessed 50% by examination and 50% by coursework (implementation exercise in Python).