School of Mathematics
Course 461 - Numerical Solution of Differential Equations I 1998-99 (Optional JS & SS Mathematics )
Lecturer: Dr J Sexton
Requirements/prerequisites: 261

Duration: 24 weeks

Number of lectures per week: 3 (including tutorials)

Assessment: Examination and Project (some years)

End-of-year Examination: One 3-hour examination

Description:

1. Elliptic Boundary Value Problems

1.1 Abstract Problems
The symmetric case. Variational inequalities
The nonsymmetric case. The Lax-Milgram lemma
1.2 Examples of elliptic boundary value problems
The Sobolev spaces Hm(W). Green's formulas
First examples of second-order boundary value problems
The elasticity problem
Examples of fourth-order problems: The biharmonic problem, the plate problem.

2. Introduction to the Finite Element Method

2.1 Basic aspects of the finite element method
The Galerkin and Ritz methods
The three basic aspects of the finite element method. Conforming finite element methods
2.2 Examples of finite elements and finite element spaces
Requirements for finite element spaces
First examples of finite elements for second order problems : n- Simplicies of type (k), (3¢)
Assembly in triangulations. The associated finite element spaces n-Rectangles of type (k). Rectangles of type (2'), (3'). Assembly in triangulations
First examples of finite elements with derivatives as degrees for freedom: Hermite n- simplicies of type (3), (3'). Assembly in triangulations
First examples of finite elements for fourth-order problems: the Argyris and Bell triangles, the Bogner-Fox-Schmit rectangle. Assembly in triangulations
2.3 General properties of finite elements and finite element spaces
Finite elements as triples (K,P,S). Basic definitions. The P-interpolation operator
Affine families of finite elements
Construction of finite element spaces Xh. Basic definitions. The Xh-interpolation operator
Finite elements of class \calC0 and \calC1
Taking into account boundary conditions. The spaces X0h and X00h
2.4 General consideration on convergence
Convergence familily of discrete problems
Céa's lemma. First consequences. Orders of convergence

3. Conforming Finite Element Methods for Second Order Problems

3.1 Interpolation theory in Sobolev spaces
The Sobolev spaces Wm,p(W). The quotient space Wk+1,p(W)/Pk(W)
Error estimates for polynomial preserving operators
Estimates of the interpolation errors |v-Pkv|m.q.k for affine families of finite elements
3.2 Application to second-order problems over polygonal domains
Estimate of the error ||u-uh||l,W
Sufficient conditions for limh® 0||u-uh||l,W = 0
Estimate of the error |u-uh|0,W. The Aubin-Nitsche lemma

Jun 10, 1998