Finite element methods for time-harmonic wave equations

Loading...
Thumbnail Image
Journal Title
Journal ISSN
Volume Title
Perustieteiden korkeakoulu | Doctoral thesis (article-based)
Checking the digitized thesis and permission for publishing
Instructions for the author
Date
2011
Major/Subject
Mcode
Degree programme
Language
en
Pages
Verkkokirja (1551 KB, 47 s.)
Series
Aalto University publication series DOCTORAL DISSERTATIONS , 88/2011
Abstract
This thesis concerns the numerical simulation of time-harmonic wave equations using the finite element method. The main difficulties in solving wave equations are the large number of unknowns and the solution of the resulting linear system. The focus of the research is in preconditioned iterative methods for solving the linear system and in the validation of the result with a posteriori error estimation. Two different solution strategies for solving the Helmholtz equation, a domain decomposition method and a preconditioned GMRES method are studied. In addition, an a posterior error estimate for the Maxwell's equations is presented. The presented domain decomposition method is based on the hybridized mixed Helmholtz equation and using a high-order, tensorial eigenbasis. The efficiency of this method is demonstrated by numerical examples. As the first step towards the mathematical analysis of the domain decomposition method, preconditioners for mixed systems are studied. This leads to a new preconditioner for the mixed Poisson problem, which allows any preconditioned for the first order finite element discretization of the Poisson problem to be used with iterative methods for the Schur complement problem. Solving the linear systems arising from the first order finite element discretization of the Helmholtz equation using the GMRES method with a Laplace, an inexact Laplace, or a two-level preconditioner is discussed. The convergence properties of the preconditioned GMRES method are analyzed by using a convergence criterion based on the field of values. A functional type a posterior error estimate is derived for simplifications of the Maxwell's equations. This estimate gives computable, guaranteed upper bounds for the discretization error.
Description
Supervising professor
Stenberg, Rolf, Prof.
Thesis advisor
Stenberg, Rolf, Prof.
Keywords
finite element method, time-harmonic wave equations, Helmholtz equation, fast solution methods
Other note
Parts
  • [Publication 1]: A. Hannukainen. Field of values analysis of preconditioners for the Helmholtz equation in lossy media. 36 pages, arXiv:1106.0424, June 2011. © 2011 by author.
  • [Publication 2]: A. Hannukainen. Continuous preconditioners for the mixed Poisson problem. 19 pages, BIT Numer. Math., published electronically, July 2011.
  • [Publication 3]: A. Hannukainen, M. Huber, J. Schöberl. A mixed hybrid finite element method for the Helmholtz equation. Journal of Modern Optics, 58, Nos. 5-6, p.424-437, 10-20 March 2011.
  • [Publication 4]: A. Hannukainen. Functional Type A Posteriori Error Estimates for Maxwell's Equations. In Numerical Mathematics and Advanced Applications, Proceedings of ENUMATH 2007, p.41-48, 2008.
Citation