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

TL;DR

This paper introduces a scalable and efficient method combining Isogeometric Analysis, deflation, and multigrid preconditioning to solve the Helmholtz equation accurately and efficiently across different dimensions and wave numbers.

Contribution

It proposes a novel approach integrating deflation with an approximate inverse of the CSLP using multigrid, enhancing scalability and reducing computational time for Helmholtz problems.

Findings

Scalable convergence with respect to wave number and discretization order.

Significant reduction in computational time compared to exact CSLP inverse.

Effective for both 1D and 2D Helmholtz problems with constant and non-constant wave numbers.

Abstract

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.