Addendum to "A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator"' [J. Comput. Appl. Math. 271 (2014) 83–99]

This communication gives an addendum to the paper Conen et al. (2014)

A Two Level Domain Decomposition Preconditionner Based on Local Dirichlet to Neumann Maps

Coarse grid correction is a key ingredient in order to have scalable domain decomposition methods. In this work we construct the coarse grid space using the low frequency modes of the subdomain DtN (Dirichlet-Neumann) maps, and apply the obtained two-level preconditioner to the linear system arising from an overlapping domain decomposition. Our method is suitable for the parallel implementation and its efficiency is demonstrated by numerical examples on problems with high heterogeneities

Towards Accuracy and Scalability: Combining Isogeometric Analysis with Deflation to Obtain Scalable Convergence for the Helmholtz Equation

Finding fast yet accurate numerical solutions to the Helmholtz equation remains a challenging task. The pollution error (i.e. the discrepancy between the numerical and analytical wave number k) requires the mesh resolution to be kept fine enough to obtain accurate solutions. A recent study showed that the use of Isogeometric Analysis (IgA) for the spatial discretization significantly reduces the pollution error. However, solving the resulting linear systems by means of a direct solver remains computationally expensive when large wave numbers or multiple dimensions are considered. An alternative lies in the use of (preconditioned) Krylov subspace methods. Recently, the use of the exact Complex Shifted Laplacian Preconditioner (CSLP) with a small complex shift has shown to lead to wave number independent convergence while obtaining more accurate numerical solutions using IgA. In this paper, we propose the use of deflation techniques combined with an approximated inverse of the CSLP using a geometric multigrid method. Numerical results obtained for both one- and two-dimensional model problems, including constant and non-constant wave numbers, show scalable convergence with respect to the wave number and approximation order p of the spatial discretization. Furthermore, when kh is kept constant, the proposed approach leads to a significant reduction of the computational time compared to the use of the exact inverse of the CSLP with a small shift

Deflation and augmentation techniques in Krylov linear solvers

Preliminary version of the book chapter entitled "Deflation and augmentation techniques in Krylov linear solvers" published in "Developments in Parallel, Distributed, Grid and Cloud Computing for Engineering", ed. Topping, B.H.V and Ivanyi, P., Saxe-Coburg Publications, Kippen, Stirlingshire, United Kingdom, ISBN 978-1-874672-62-3, p. 249-275, 2013In this paper we present deflation and augmentation techniques that have been designed to accelerate the convergence of Krylov subspace methods for the solution of linear systems of equations. We review numerical approaches both for linear systems with a non-Hermitian coefficient matrix, mainly within the Arnoldi framework, and for Hermitian positive definite problems with the conjugate gradient method.Dans ce rapport nous présentons des techniques de déflation et d'augmentation qui ont été développées pour accélérer la convergence des méthodes de Krylov pour la solution de systémes d'équations linéaires. Nous passons en revue des approches pour des systémes linéaires dont les matrices sont non-hermitiennes, principalement dans le contexte de la méthode d'Arnoldi, et pour des matrices hermitiennes définies positives avec la méthode du gradient conjugué