The algebraic preconditioner for saddle point systems based on the properties of a PDE

Bachelor thesis (2024)

Authors

J. van der Wagt Electrical Engineering, Mathematics and Computer Science

Contributors

Carolina Urzúa-Torres Numerical Analysis - EEMCS (mentor)
H.M. Schuttelaars Mathematical Physics - EEMCS (mentor)
M.E. Kootte Mathematical Physics - EEMCS (mentor)
Faculty
Electrical Engineering, Mathematics and Computer Science
Preconditioning Saddle point system Finite difference algorithm
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Published Date

09-07-2024

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Abstract

Differential equations are commonly solved by, first, discretising the domain and then using an iterative method on the resulting system of equations. Refining the mesh gives a more accurate solution. However, the iterative method does not necessarily converge quickly to a good approximate solution anymore. To deal with this issue, we can add a preconditioner to the system. A good preconditioner enhances the speed of convergence.

Two ways of finding a good preconditioner are known. The first uses the properties of the matrix of the system, that is an algebraic preconditioner. The second uses the properties of the differential equation that give rise to the system, which is an operator preconditioner. This thesis makes a connection between both types of preconditioning.

First, we discretise a differential equation and perform numerical test on the system to see if the
algebraic preconditioning works. Then, the domains on which the matrix and preconditioner act are defined in terms of the differential equations.

We conclude that the matrix containing the differential operators acts on H(div, Ω) × H1(Ω) and the preconditioner acts on [L2(Ω)]^2× H2(Ω). So we need to constrain the domains in such a way that the operators both act on the same domain: that is, H(div, Ω) × H2(Ω). If the function is in this domain,then we know that the algebraic preconditioner is an effective preconditioner.