Towards accuracy and scalability: Combining Isogeometric Analysis with deflation to obtain scalable convergence for the Helmholtz equation

Dwarka, V.N.S.R.; Tielen, R.P.W.M.; Möller, M.; Vuik, Cornelis

doi:10.1016/j.cma.2021.113694

Towards accuracy and scalability

Title

Towards accuracy and scalability: Combining Isogeometric Analysis with deflation to obtain scalable convergence for the Helmholtz equation

Author

Dwarka, V.N.S.R. (TU Delft Numerical Analysis)
Tielen, R.P.W.M. (TU Delft Numerical Analysis)
Möller, M. (TU Delft Numerical Analysis)
Vuik, Cornelis (TU Delft Numerical Analysis)

Date

2021

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 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 multigrid-approximated or exact inverse of the CSLP with a small shift, in particular for three-dimensional model problems.

Subject

Deflation
GMRES
Helmholtz
Isogeometric Analysis
Numerical dispersion
Pollution

To reference this document use:

http://resolver.tudelft.nl/uuid:eef4dcc4-57af-4e79-93a3-25c708db32d7

DOI

https://doi.org/10.1016/j.cma.2021.113694

ISSN

0045-7825

Source

Computer Methods in Applied Mechanics and Engineering, 377

Part of collection

Institutional Repository