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COMP60092: Computational Finite Element Methods (2009-2010)

This is an archived syllabus from 2009-2010

Computational Finite Element Methods
Level: 6
Credit rating: 15
Pre-requisites: No Pre-requisites
Co-requisites: No Co-requisites
Lecturers: Milan Mihajlovic
Course lecturer: Milan Mihajlovic

Additional staff: view all staff
Timetable
SemesterEventLocationDayTimeGroup
Sem 2 w19-23,25-26,30-33 Lecture 2.15 Tue 09:00 - 13:00 -
Assessment Breakdown
Exam: 35%
Coursework: 65%
Lab: 0%

Introduction

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 [1], will be used as a testbed throughout the course.

Aims

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.

Learning Outcomes

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 outcomes

Learning outcomes (1) and (2) are assessed by examination, in the laboratory and via the mini-project

Contribution to Programme Learning Outcomes

A1, A2, B2, C1 and C3

Syllabus

Introductory remarks

Review of the numerical algorithms that are the building blocks of the finite element method. Lagrange and Hermite interpolation, Gauss quadrature rules. [1]

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. [4]

Finite element method for the Poisson equation

Galerkin'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. [7]

Finite element methods for the convection-diffusion equation

Applications 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. [6]

The heat equation

Linear multistep methods for the solution of initial value problems (IVPs). Backward differentiation formulas (BDFs). Predictor-corrector schemes for adaptive time stepping. [3]

Mixed finite element methods

Saddle-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. [9]

Reading List

Core Text
Title: Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics
Author: Elman, Howard, David Silvester, and Andy Wathen
ISBN: 9780198528685
Publisher: Oxford Science Publications
Edition:
Year: 2005


Supplementary Text
Title: Computational differential equations
Author: Eriksson, Kenneth et al.
ISBN: 0521563127
Publisher: Cambridge University Press
Edition:
Year: 1996


Supplementary Text
Title: Multigrid tutorial (2nd edition)
Author: Briggs, William L., Van Emden Henson, and Steve F. McCormick
ISBN: 0898714621
Publisher: SIAM (Society for Industrial and Applied Mathematics)
Edition: 2nd
Year: 2000