Current postgraduate taught students
COMP60092: Computational Finite Element Methods (2007-2008)
This module is concerned with the practical computational issues of the numerical methods used for the solution of partial differential equations. Particular attention is devoted to the finite element method for the spatial approximation of stationary problems, adaptive predictor-corrector schemes, based on linear multistep methods, for the solution of transient problems, and efficient iterative methods for the solution of large, sparse systems of linear equations which arise in this context. In order to achieve practical understanding of the material, the IFISS package written in Matlab, which accompanies the textbook , will be used as a testbed throughout the course.
To give an understanding of practical computational issues of the finite element apprximation methods. To provide the students with technical knowledge at implementation level needed to solve practical PDE problems.
On successful completion of the module, students will
understand the concept of Galerkin's approximation method and the finite element method as its particular instance based on piecewise polynomial approximation;
understand the methodology for the integration of time-dependent problems, based on adaptive predictor-corrector methods;
have an appreciacion of the sophisticated iterative methods and preconditioners, based on multigrid for the solution of linear systems which arise in the finite element approximations.
have practical knowledge at implementation level of the finite element method and iterative solution techniques for large sparse linear systems.
Assessment of Learning outcomesLearning outcomes (1) and (2) are assessed by examination, in the laboratory and via the mini-project
Contribution to Programme Learning OutcomesA1, A2, B2, C1 and C3
Introductory remarksReview of the numerical algorithms that are the building blocks of the finite element method. Lagrange and Hermite interpolation, Gauss quadrature rules. 
Finite element method for ordinary differential equations (ODEs)Galerkin's approximation method. Choice of test and solution spaces. Different types of boundary conditions. Piecewise linear approximation. Interelement continuity of the solution. Local and global basis functions. Finite element basis set. Assembly of the Galerkin matrix. Solution of tridiagonal linear systems. Piecewise quadratic elements. 
Finite element method for the Poisson equationGalerkin's approximation method. Choice of test and solution spaces. Different types of boundary conditions. Finite element approximation on tensor product grids. Piecewise bilinear and biquadratic approximation. Local and global bilinear basis functions. Finite element approximation on non-structured triangular grids. Piecewise linear approximation. Local and global linear basis functions. Area coordinates. Assembly of the Galerkin matrix. Piecewise quadratic elements. Basic problems and inefficiency of the fixed point iterations in the context of finite element linear systems. Fourier decomposition of the solution error. Smoothing property of the fixed-point iterations. Recursive (multilevel) application of fixed-point iterations and the V-cycle of multigrid. 
Finite element methods for the convection-diffusion equationApplications of the convection-diffusion equation. Relative measurement of the convection and the diffusion. Peclet number. Galerkin's approximation method. The streamline diffusion method. Adaptive Galerkin's approximation method. 
The heat equationLinear multistep methods for the solution of initial value problems (IVPs). Backward differentiation formulas (BDFs). Predictor-corrector schemes for adaptive time stepping. 
Mixed finite element methodsSaddle-point problems. The Stokes equations. Conforming finite element spaces. Linear algebra interpretation of the inf-sup stability condition. Efficient iterative solution and block preconditioning (based on multigrid) of the discrete Stokes systems. 
Core TextTitle: Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics
Author: Elman, Howard, David Silvester, and Andy Wathen
Publisher: Oxford Science Publications
Supplementary TextTitle: Computational differential equations
Author: Eriksson, Kenneth et al.
Publisher: Cambridge University Press
Supplementary TextTitle: Multigrid tutorial (2nd edition)
Author: Briggs, William L., Van Emden Henson, and Steve F. McCormick
Publisher: SIAM (Society for Industrial and Applied Mathematics)