Synopsis

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 provide an introduction to their mathematical theory, with special emphasis on theoretical questions such as accuracy, reliability and adaptivity; practical issues concerning the development of efficient finite element algorithms will also be discussed.

Syllabus:

Elements of function spaces. Elliptic boundary value problems: existence, uniqueness and regularity of weak solutions.

Finite element methods: Galerkin orthogonality and Cea's lemma. Piecewise polynomial approximation in Sobolev spaces. Optimal error bounds in the energy norm. Variational crimes.

The Aubin-Nitsche duality argument. Superapproximation properties in mesh-dependent norms. A posteriori error analysis by duality: reliability, efficiency and adaptivity.

Finite element approximation of initial boundary value problems: Stability and error analysis.

Prerequisites:

While no formal prerequisites are assumed, students who take this course will find it helpful to attend the Michaelmas Term lecture course on Function Spaces for Applications.

Reading List

• S. Brenner & R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, Second Edition 2002 [Chapters 0,1,2,3; Chapter 4: Secs. 4.1--4.4, Chapter 5: Secs. 5.1--5.7].

• K. Eriksson, D. Estep, P. Hansbo, & C. Johnson, Computational Differential Equations. CUP, 1996. [Chapters 5, 6, 8, 14 -- 17].

• 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].

• E. Suli, Finite Element Methods for Partial Differential Equations. Oxford University Computing Laboratory, 2001.