Courses
CA-MATH-804: Numerical Analysis
Course Details

CA-MATH-804: Numerical Analysis

Lectures

Lectures take place: 11:15-12:30 Tuesday Thursday (East Hall 8) and Fridays (East Hall 4), from 1 February until 13 May 2022.

Content and Educational Aims

The module is an introduction to the analysis of basic classes of numerical algorithms used in large-scale scientific computing.

It introduces the fundamental notions and concepts of numerical mathematics.

Then, successively, iterative solvers, interpolation, and quadrature are discussed and analyzed.

They serve as the core numerical building blocks for an introduction to the finite element method (FEM) as one of the modern numerical techniques widely used in engineering applications and theoretical physics.

Text Books

  • “Numerical Mathematics” - Quarteroni, Sacco and Saleri (2007)
  • “A First Course in the Numerical Analysis of Differential Equations” - Iserles (2012)

The following are also useful:

  • “An Introduction to the Conjugate Gradient Method Without the Agonizing Pain” - Shewchuk (1994). Also see [here]
  • “An Introduction to Computational Stochastic PDEs” - Lord, Powell and Shardlow (2014). [Chapters 1 & 2]

Course Outline

The following topics will be covered:

Part 1: Foundations

  • Principles of numerical mathematics: well-posedness, stability, robustness, condition, and consistency
  • Equivalence theorem of Lax-Richtmyer for exemplary problems
  • Types of error analysis (forward, backward, a priori, and a posteriori)
  • Sources of errors (modelling, data, discretization, rounding, and truncation)
  • Foundations of matrix analysis: vector norms and matrix norms and compatible/consistent norms
  • Stability analysis for linear systems: the condition number of a matrix, forward/backward a priori analysis, and the convergence of iterative methods
  • Iterative methods: gradient descent and conjugate gradient method

Part 2: Interpolation and Numerical Integration

  • Review of Lagrange interpolation, error estimates, drawbacks, Runge’s counterexample the stability of polynomial interpolation, piecewise Lagrange interpolation, and extensions to the multi-dimensional case
  • Quadrature formulas: interpolatory quadrature, error estimates, Gauss quadrature, degree of exactness, and extensions to the multi-dimensional case

Part 3: Numerical Solutions of Differential Equations

  • Finite difference methods (FDM), stability and convergence analysis for FDM and error estimates for FDM
  • The notion of a weak solution
  • The Galerkin method and the Finite Element Method (FEM)
  • Error estimates for FEM: Céa’s lemma and approximate estimates
  • Particular examples in 1D and 2D and linear and quadratic shape functions

Assessment
Examination Type: Module examination
Assessment Type: Written examination
Scope: All intended learning outcomes of this module
Duration: 120 min
Weight: 100%

By submitting homework you can improve this grade by up to 0.66 points. More details about homework will be announced in class.