Deflation of the Finite Pointset Method
(2014) In Master's Theses in Mathematical Sciences FMN820 20141Mathematics (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 zerovector. 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)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/studentpapers/record/4496677
 author
 Pålsson, Sara ^{LU}
 supervisor

 Gustaf Söderlind ^{LU}
 Jörg Kuhnert
 organization
 course
 FMN820 20141
 year
 2014
 type
 H2  Master's Degree (Two Years)
 subject
 keywords
 Finite Pointset Method, Deflation
 publication/series
 Master's Theses in Mathematical Sciences
 report number
 LUTFMA30292014
 ISSN
 14046342
 other publication id
 2012:E32
 language
 English
 id
 4496677
 date added to LUP
 20140626 11:41:13
 date last changed
 20151214 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 zerovector. 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 = {{14046342}}, language = {{eng}}, note = {{Student Paper}}, series = {{Master's Theses in Mathematical Sciences}}, title = {{Deflation of the Finite Pointset Method}}, year = {{2014}}, }