Course Syllabus for

# Iterative Solution of Large Scale Systems in Scientific Computing Iterativ lösning av storskaliga system i beräkningsteknik

## FMN020F, 7.5 credits

Valid from: Spring 2016
Decided by: FN1/Anders Gustafsson
Date of establishment: 2015-11-13

## General Information

Division: Numerical Analysis
Course type: Third-cycle course
Teaching language: English

## Aim

A core problem in Scientific Computing is the solution of nonlinear and linear systems. These arise in the solution of boundary value problems, stiff ordinary differential equations and in optimization. Particular difficulties appear when the systems are large, meaning millions of unknowns. This is often the case when discretizing partial differential equations which model important phenomenas in science and technology. Due to the size of the systems they may only be solved using iterative methods. The aim of this course is to teach modern methods for the solution of such systems. The course is a direct follow up of the course FMNN10 Numerical Methods for Differential Equations, and expands the postgraduate student's toolbox for calculating approximative solutions of partial differential equations.

## Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• understand basic iterative methods for linear and nonlinear equations, and understand their mathematical differences
• understand the framwork of Jacobian-free Newton-Krylov methods
• understand multigrid methods for model problems.

Competences and Skills

For a passing grade the doctoral student must

• be able to implement an inexact Jacobian-free Newton-Krylov method
• be able to implement a multigrid method for model problems
• be able to apply basic iterative solvers in a computer program.

Judgement and Approach

For a passing grade the doctoral student must be able to decide, given information about a nonlinear or linear system, which solver to use and which not to.

## Course Contents

Where do large scale linear and nonlinear systems arise in Scientific Computing? Speed of convergence Termination criteria Fixed Point mehtods and convergence theory Newton's method, its convergence theory and its problems Inexact Newton's method and its convergence theory Methods of Newton type and convergence theory Linear systems Krylov subspace methods and GMRES - the Generalized Minimal RESidual method Preconditioning GMRES Jacobian-free Newton-Krylov methods Multigrid methods in one and two dimensions Multigrid methods for nonstandard equations and for nonlinear systems

## Course Literature

• Kelley, C. T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, 1995. ISBN 9780898713527.
• Wesseling, P.: An Introduction to Multigrid Methods. R T Edwards, 2004. ISBN 9781930217089.

## Instruction Details

Types of instruction: Lectures, exercises, project

## Examination Details

Examination formats: Oral exam, written report