Abstract
This thesis concerns the numerical simulation of timeharmonic 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 highorder, 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 twolevel 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.
Translated title of the contribution  Finite element methods for timeharmonic wave equations 

Original language  English 
Qualification  Doctor's degree 
Awarding Institution 

Supervisors/Advisors 

Publisher  
Print ISBNs  9789526042961 
Electronic ISBNs  9789526042978 
Publication status  Published  2011 
MoE publication type  G5 Doctoral dissertation (article) 
Keywords
 finite element method
 timeharmonic wave equations
 Helmholtz equation
 fast solution methods
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Hannukainen, A. (2011). Finite element methods for timeharmonic wave equations. Aalto University. http://urn.fi/URN:ISBN:9789526042978