(Created 2011-09-01.)
 MULTIGRID METHODS FOR DIFFERENTIAL EQUATIONS FMNN15
Credits: 4. Grading scale: UG. Cycle: A (Second Cycle). Main field: Technology. Language of instruction: The course will be given in English on demand. FMNN15 overlaps following cours/es: FMN130. Compulsory for: Pi3. Optional for: F4. Course coordinator: Gustaf Söderlind,, Gustaf.Soderlind@na.lu.se, and Director of studies Anders Holst, Anders.Holst@math.lth.se, Numerical Analysis. Recommended prerequisits: FMNN10 Numerical methods for differential equations, or equivalent. Assessment: Computer project with report. The project aims at the solution of a large computational problem (e.g. determining the electrical potential in a three dimensional region) using elementary multigrid methods. Home page: http://www.maths.lth.se/na/courses/FMNN15/.

Aim
Many important phenomena in science and engineering are described by partial differential equations. When these equations are solved numerically one uses discretization methods, which give rise to (often enormously) large systems of equations. It is not unusual that these systems have several million unknowns. Due to the size of the systems it is necessary to use iterative methods, with multigrid methods belonging to the most efficient techniques.

The course builds directly on FMNN10 Numerical Solution of Differential Equations, and is focused on multigrid methods for elliptic equations. The aim is to give an elementary introduction to multigrid, starting from the self-adjoint two-point boundary value problems studied in FMNN10. Then the technique is applied to more general elliptic equations, and different variants such as V- and W-iterations are used.

Knowledge and understanding
For a passing grade the student must

- understand in what contexts multigrid methods can be applied, as well as their connection to the numerical solution of elliptic partial differential equations.

- understand the ideas behind different variants of multigrid, such as V- and W-iterations.

- have a basic understanding of the convergence of multigrid methods.

Skills and abilities
For a passing grade the student must

- be able to implement a simple multigrid method.

- be able to interpret the convergence of multigrid iterations in practice and assess the results.

- be able to solve simple elliptic applied problems with a multigrid method.

Judgement and approach
For a passing grade the student must

- be a able to describe method, implementation and results in a well-structured manner, using correct terminology and clear logic.

Contents

• Convergence of linear iterations in relation to the spectral properties of the differential or difference operator.

• Multigrid iteration, V cycles, W cycles.

• Poisson's equation with multigrid, preconditioning.

Literature
Briggs, W.L. A Multigrid tutorial, 2nd ed. SIAM 2000. ISBN 978-0898714623.
Supplementary material from the department.