(Created 2009-08-11.)
 MULTIGRID METHODS FOR DIFFERENTIAL EQUATIONS FMNN15

Higher education credits: 4. Grading scale: TH. Level: A (Second level). Language of instruction: The course will be given in English on demand. FMNN15 overlap following cours/es: FMN130 och FMN130. Compulsory for: Pi3. Course coordinator: Gustaf Söderlind, Gustaf.Soderlind@na.lu.se, Numerisk analys. Recommended prerequisits: Having participated fully in FMNN10 Numerical methods for differential equations, or equivalent. Assessment: Computer project with report.

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. These may often 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 FMNxxx. 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 of 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

with correct terminology, well structured and with clear logic be able to describe method, implementation and results.

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. SIAM 1987. ISBN 0-89871-221-1.
Alternative:
Iserles, A. A first course in the Numerical Analysis of Differential Equations. Cambridge UP. ISBN 0-521-55655-4.
Supplementary material from the department.