MA4N6 Finite-Element methods for PDEs

Introductory description

The module will introduce students to the analysis and numerical approximation of variational solutions of partial differential equations (PDEs). Many mathematical models are based on PDEs but their complexity generally make it impossible to solve these explicitly. Existence and uniqueness of solutions to these problems can be shown based on their variational formulation but this on the face of it, does not provide a practical way of evaluating solutions. In this module we will discuss how the variational formulation can be used to construct so called Galerkin approximations. The most commonly used form of these is the finite element method. We will discuss both implementational and analytical aspects of these method in this module.

We will start with an overview of variational formulations of PDEs and the relevant function spaces so that MA4A2 (Advanced PDEs) is not a prerequisite. Some results from functional analysis will only be recalled, e.g., Riesz representation and Lax-Milgram theorems.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

This is an indicative module outline only to give an indication of the sort of topics that may be covered.

• Sobolev spaces and their main properties in connection with variational formulations of PDEs
• Galerkin approximations of variational problems
• Introduction to implementational aspects of finite element methods
• Approximation theory and a-priori error analysis
• Derivation of a-posteriori error estimators
• How the concepts are used to solve non-linear and time dependent problems

Learning outcomes

By the end of the module, students should be able to:

• have an overview of Sobolev spaces and their main properties
• understand the fundamental concepts of Galerkin approximations to weak solutions of linear PDEs
• implement finite-element methods to compute approximations
• carry out a-priori error analysis based on the Bramble-Hilbert theory
• understand the difference between a-priori and a-posteriori analysis and how to obtain a-posteriori error estimators.
• understand how the Galerkin method can be used to solve non-linear and time dependent problems

Indicative reading list

• Lawrence C Evans Partial Differential Equations AMS
• Dietrich Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics (Cambridge University Press
• S Brenner and L Ridgeway Scott, The mathematical theory of finite element methods Springer Texts in Applied Mathematics Volume 15.
• Michael Reed and Barry Simon An Introduction to Partial Differential Equations Springer.

Subject specific skills

After completing this module, students will be able to:

• Recall the definition of the Sobolev spaces and formulate notions of weak solutions for linear elliptic PDEs.
• approximate variational problems using Galerkin methods
• implement finite-element methods
• prove a-priori and a-posteriori error estimates for finite element approximations
• use the finite-element method to also solve non-linear and time dependent PDEs

Transferable skills

Students will have a detailed understanding how PDEs are formulated in variational form and how this formulation can be used to construct approximations. They will have seen both theoretical and practical aspects of one of the most widely used numerical methods for solving PDEs both in academia and industry. They will have obtained a foundation for further study at graduate level, and will be able to apply the techniques they have learned in pure mathematics, the applied sciences and in many branches of industry.