# C6.4 Finite Element Method for PDEs - Material for the year 2019-2020

No formal prerequisites are assumed. The course builds on elementary calculus, analysis and linear algebra and, of course, requires some acquaintance with partial differential equations such as the material covered in the Prelims Multivariable Calculus course, in particular the Divergence Theorem. Part A Numerical Analysis would be helpful but is certainly not essential. Function Space material will be introduced in the course as needed.

16 lectures

### Assessment type:

- Written Examination

Computational algorithms are now widely used to predict and describe physical and other systems. Underlying such applications as weather forecasting, civil engineering (design of structures) and medical scanning are numerical methods which approximately solve partial differential equation problems. This course gives a mathematical introduction to one of the more widely used methods: the finite element method.

Finite element methods represent a powerful and general class of techniques for the approximate solution of partial differential equations. The aim of this course is to introduce these methods for boundary value problems for the Poisson and related elliptic partial differential equations.

Attention will be paid to the formulation, the mathematical analysis and the implementation of these methods.

- H. Elman, D. Silvester & A. Wathen,
*Finite Elements and Fast Iterative Solvers.*Second edition. OUP, 2014. [Mainly Chapters 1 and 3]. - H. Elman, D. Silvester & A. Wathen,
*Finite Elements and Fast Iterative Solvers.*OUP, 2005. [Mainly Chapters 1 and 5].

or

- S.C. Brenner & L.R. Scott,
*The Mathematical Theory of Finite Element Methods.*Springer, 2nd edition, 2002. [Chapters 0,1,2,3; Chapter 4: Secs. 4.1-4.4, Chapter 5: Secs. 5.1-5.7]. - C. Johnson,
*Numerical Solution of Partial Differential Equations by the Finite Element Method.*CUP, 1990. [Chapters 1-4; Chapter 8:Secs. 8.1-8.4.2; Chapter 9: Secs. 9.1-9.5]. - Typed lecture notes covering a previous version of the entire course (and more) are available from the course material webpage:

Endre Süli,*Finite Element Methods for Partial Differential Equations.*Mathematical Institute, University of Oxford, 2011 - Some of the introductory material is covered in:

Endre Süli & David Mayers,*An Introduction to Numerical Analysis*,CUP 2003; Second Printing 2006. [Chapter 11 and in particular Chapter 14].