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


  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