The construction of optimal solvers for high frequency Helmholtz-type equations is highly problematic. After discretisation of the previous equations by a finite element method,the underlying linear systems are usually large and difficult to solve both by direct and iterative methods. Domain decomposition methods are hybrid methods in the sense that they use an iterative coupling of smaller problems that are solved by direct methods,and rely on the splitting the global problem into local problems on smaller subdomains. These methods can be used as iterative solvers but also as preconditioners in a Krylov type method. That is the reason why transmission conditions between subdomains are very important. In this manuscript, we start by an overview of main domain decomposition methods and focus first on their use as preconditioners. Then we consider these methods from an iterative point of view and perform a convergence study of non-overlapping and overlapping Schwarz methods with Dirichlet and Robin interface conditions, by analysing their behaviour and conclude on their convergence properties which prove to be very poor when used as solvers. The theoretical findings are illustrated by numerical results. Then we present more sophisticated methods, namely the optimised Schwarz algorithms,which use more effective transmission conditions depending on some parameters which are solutions of min-max problems. The Schwarz preconditioners defined previously were one-level, meaning only the information from the neighbouring domains is used. This has the undesired consequence that the number of iterations needed to reach convergence increases with the number of subdomains. For this reason we have tested numerically two-level preconditioners,based on a coarse grid correction, this very simple idea giving promising results.
|Date of Award||29 Mar 2019|
- University Of Strathclyde
|Sponsors||University of Strathclyde|
|Supervisor||Victorita Dolean Maini (Supervisor) & Stella Pytharouli (Supervisor)|