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Deflation of the Finite Pointset Method

(2014) In Master's Theses in Mathematical Sciences FMN820 20141
Mathematics (Faculty of Engineering)
Abstract
In this thesis a deflation method for the Finite Pointset Method (FPM) is presented. FPM is a particle method based on Lagrangian coordinates to solve problems in fluid dynamics. A strong formulation of the occuring differential equations is produced by FPM, and the linear system of equations obtained by an implicit approach is solved by an iterative method such as BiCGSTAB. To improve the convergence rate of BiCGSTAB, the computational domain is divided into a number of deflation cells and a projection between the deflated domain and the original domain is constructed with the help of different ansatz functions, either constant, linear or quadratic. Also, the Moore Penrose pseudoinverse of the projection is computed. Applying the... (More)
In this thesis a deflation method for the Finite Pointset Method (FPM) is presented. FPM is a particle method based on Lagrangian coordinates to solve problems in fluid dynamics. A strong formulation of the occuring differential equations is produced by FPM, and the linear system of equations obtained by an implicit approach is solved by an iterative method such as BiCGSTAB. To improve the convergence rate of BiCGSTAB, the computational domain is divided into a number of deflation cells and a projection between the deflated domain and the original domain is constructed with the help of different ansatz functions, either constant, linear or quadratic. Also, the Moore Penrose pseudoinverse of the projection is computed. Applying the projection and restriction to the linear FPM system, a deflated system is obtained which can easily be solved with a direct method. The deflated solution is then projected onto the full domain.

The deflation is tested for a number of test cases in one and two dimensions. Constant ansatz functions provide acceptable results for Dirichlet problems, but give big errors when deflating problems with mixed boundary conditions. Linear ansatz functions provide good approximations which converge to the exact solution as the number of deflation cells increases. Quadratic ansatz functions provide deflated solutions as good as the exact solutions for all test cases but are computationally expensive. The BiCGSTAB convergence rate is improved when using a deflated solution as initial guess, compared to using the zero-vector. The size of the improvement varies between which ansatz functions are used.

Overall, the proposed method provides an increased convergence rate in the BiCGSTAB algorithm for FPM. However, the computational effort in the deflation process should also be taken into account. (Less)
author
supervisor
organization
course
FMN820 20141
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Finite Pointset Method, Deflation
publication/series
Master's Theses in Mathematical Sciences
report number
LUTFMA-3029-2014
ISSN
1404-6342
other publication id
2012:E32
language
English
id
4496677
2014-06-26 11:41:13
date last changed
2015-12-14 13:32:15
```@misc{4496677,
abstract     = {{In this thesis a deflation method for the Finite Pointset Method (FPM) is presented. FPM is a particle method based on Lagrangian coordinates to solve problems in fluid dynamics. A strong formulation of the occuring differential equations is produced by FPM, and the linear system of equations obtained by an implicit approach is solved by an iterative method such as BiCGSTAB. To improve the convergence rate of BiCGSTAB, the computational domain is divided into a number of deflation cells and a projection between the deflated domain and the original domain is constructed with the help of different ansatz functions, either constant, linear or quadratic. Also, the Moore Penrose pseudoinverse of the projection is computed. Applying the projection and restriction to the linear FPM system, a deflated system is obtained which can easily be solved with a direct method. The deflated solution is then projected onto the full domain.

The deflation is tested for a number of test cases in one and two dimensions. Constant ansatz functions provide acceptable results for Dirichlet problems, but give big errors when deflating problems with mixed boundary conditions. Linear ansatz functions provide good approximations which converge to the exact solution as the number of deflation cells increases. Quadratic ansatz functions provide deflated solutions as good as the exact solutions for all test cases but are computationally expensive. The BiCGSTAB convergence rate is improved when using a deflated solution as initial guess, compared to using the zero-vector. The size of the improvement varies between which ansatz functions are used.

Overall, the proposed method provides an increased convergence rate in the BiCGSTAB algorithm for FPM. However, the computational effort in the deflation process should also be taken into account.}},
author       = {{Pålsson, Sara}},
issn         = {{1404-6342}},
language     = {{eng}},
note         = {{Student Paper}},
series       = {{Master's Theses in Mathematical Sciences}},
title        = {{Deflation of the Finite Pointset Method}},
year         = {{2014}},
}

```