Inexact subdomain solves using deflated GMRES for Helmholtz problems

We examine the use of a two-level deflation preconditioner combined with GMRES to locally solve the subdomain systems arising from applying domain decomposition methods to Helmholtz problems. Our results show that the direct solution method can be replaced with an iterative approach. This will be particularly important when solving large 3D high-frequency problems as subdomain problems can be too large for direct inversion or otherwise become inefficient. We additionally show that, even with a relatively low tolerance, inexact solution of the subdomain systems does not lead to a drastic increase in the number of outer iterations. As a result, it is promising that a combination of a two-level domain decomposition preconditioner with inexact subdomain solves could provide more economical and memory efficient numerical solutions to large-scale Helmholtz problems

Inexact Subdomain Solves Using Deflated GMRES for Helmholtz Problems

In recent years, domain decomposition based preconditioners have become popular tools to solve the Helmholtz equation. Notorious for causing a variety of convergence issues, the Helmholtz equation remains a challenging PDE to solve numerically. Even for simple model problems, the resulting linear system after discretisation becomes indefinite and tailored iterative solvers are required to obtain the numerical solution efficiently. At the same time, the mesh must be kept fine enough in order to prevent numerical dispersion ‘polluting’ the solution [4]. This leads to very large linear systems, further amplifying the need to develop economical solver methodologies.Numerical Analysi