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|Lecturer:||Dr Niall Madden
|Lectures:||Monday at 11.00
Tuesday at 11.00
Wednesday at 12.00
Classes take place in ADB-1020.
|Starts:||Monday, Sep 8, at 11.00.|
|Lecture summaries:||see below.|
|Credits:||10 ECTS (a 5 ECTS variant may be available to students in structured PhD programmes).|
|Overview:||This module is concerned with the mathematics of algorithms for
the numerical solution of differential equations, mainly by finite
It is aimed at graduate students (taught Masters and structured PhD programmes) in mathematical and computational sciences.
Except in very rare cases, it is not possible to solve differential equations by apolitical methods. The best one can do is generate approximate solutions using numerical algorithms. In this module, we mainly focus on the use of finite difference methods for ordinary and partial differential equations. The construction/derivation of these methods will be our initial focus. However, most of the effort will go into analysing the methods, and establishing rigorous error bounds. We'll also study their implementation in Matlab, though this will be mainly in support of our theoretical studies.
Through the semester, I'll use various texts. As I introduce a text, the details will be added below. Some of these are published by SIAM and are freely available to you on-line through NUI Galway's subscription to SIAM e-books.
These are extremely brief accounts of what was discussed in class at each lecture. Please refer to your lecture notes.
|Lecture 1:||Welcome to the Masters Programme, and to MA500-1||11am, Monday, 08/09/14|
|Lecture 2:||Approximation of derivatives||11am, Tuesday, 09/09/14|
|Lecture 3:||Taylor's series||12pm, Wednesday, 10/09/14|
|Lecture 4:||Introduction to finite difference methods||11am, Monday, 15/09/14|
|Lecture 5:||Towards error analysis||11am, Tuesday, 16/09/14||Code: SimpleFD1.m|
|Lecture 6:||Error analysis: Part 1||12pm, Wednesday, 17/09/14|
|Lecture 7:||Norms (i)||11am, Monday, 22/09/14||Lecture notes|
|Lecture 8:||Norms (ii): norms of matrices||11am, Tuesday, 23/09/14||Lecture notes|
|Class 9:||More norms||12pm, Wednesday, 24/09/14|
|Class 10:||Finite differences in Matlab (1).||11am, Thursday, 25/09/14||Class notes|
|Class 11: ||Consistency + Stability
⇒ Convergence ||11am, Monday, 29/09/14
||Class 12: ||Convergence in the 2-norm
||11am, Tuesday, 30/09/14
||Class 13: ||Convergence in the max-norm
||12pm, Wednesday, 01/10/14
||Class 14: ||Maximum principles for
differential equations ||11am, Monday, 06/10/14
||Class 15: ||Maximum principles for
difference equations ||11am, Tuesday, 07/10/14
||Class 16: ||Neumann boundary conditions (1) ||12pm, Wednesday, 08/10/14
||Class 17: ||Neumann boundary conditions (2) ||11am, Monday, 13/10/14
||Class 18: ||General boundary value problems ||11am, Tuesday, 14/10/14
||Class 19: ||More on general BVPs ||12pm, Wednesday, 15/10/14
||Class 20: ||Elliptic BVPs in two
||Class 21: ||A simple FEM for the Poisson problem ||11am,
||Class 22: ||Ordering the unknowns in a 2D
FEM ||12pm, Wednesday, 22/10/14
||Class 23: ||Expressing finite difference
methods using Kronecker products ||11am, Tuesday, 28/10/14
||Problem Set 2 ||Brief details of each class.