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Details for the Course Syllabus for Course FMN020F valid from Spring 2016

General
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.
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

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.
Types of Instruction
• Lectures
• Exercises
• Project
Examination Formats
• Oral exam
• Written report
• Failed, pass
Assumed Prior Knowledge
• FMNN10 Numerical Methods for Differential Equations.
Selection Criteria
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.
Further Information
Course code
• FMN020F