Iterative solution of Helmholtz problem with high-order isogeometric analysis and finite element method at mid-range frequencies

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

“…In this work, we focus on the combination of IgA discretized linear systems with a state-of-the-art iterative solver using deflation and a geometric multigrid method. In particular, we extend the line of research set out by 15 , where it was shown that the use of IgA reduces the pollution error significantly compared to −order FEM. The authors have shown that the use of the exact inverse of the CSLP preconditioner with a small complex shift, yields wave number independent convergence for moderate values of .…”

“…We start by studying the pollution error for our one-dimensional model problem when adopting high-order B-spline basis functions for the spatial discretization. In 15 , a detailed first application of IgA discretizations for Helmholtz problems has been given. We therefore only show the pollution reduction for the model problems used in this paper.…”

“…Thus, while the use of IgA for Helmholtz-type problems becomes more established, the process of solving the underlying discretized systems remained fairly untouched. Until recently, a study by Diwan et al 15 covered this for the Helmholtz equation and researched the use of IgA together with an iterative solver. There, the resulting linear systems are solved using the Generalized Minimum Residual Krylov method (GMRES) preconditioned with the Complex Shifted Laplacian Preconditioner (CSLP) using a small complex shift.…”

“…Consequently, our aim in this paper is to extend the research direction set out in 15,14 , by combining state-of-the-art iterative solvers with IgA discretization techniques to obtain both accurate and computationally efficient numerical solutions. In particular, we propose the use of deflation techniques combined with an approximated inverse of the CSLP using multigrid to obtain scalable and faster convergence with respect to the wave number and the order .…”

Isogeometric simulation of acoustic radiation

“…Hence, another difficulty for the numerical solution of the Helmholtz equation is that for k sufficiently large, the coefficient matrix is indefinite and non-normal. As a consequence, iterative methods to solve the corresponding linear systems behave extremely bad if the system is not preconditioned [14], [11]. To face this problem researches have proposed several preconditioners, such as multigrid methods with Krylov smoothers, domain decomposition, and complex shifted Laplacian preconditioner.…”