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 Partial Differential Equations for Pure and Applied Mathematicians.

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. Süli, Finite Element Methods for Partial Differential Equations. Oxford University Computing Laboratory, 2001.