Towards accuracy and scalability

Combining Isogeometric Analysis with deflation to obtain scalable convergence for the Helmholtz equation

Journal Article (2021)
Author(s)

V. Dwarka (TU Delft - Numerical Analysis)

Roel Tielen (TU Delft - Numerical Analysis)

M Möller (TU Delft - Numerical Analysis)

C. Vuik (TU Delft - Numerical Analysis)

Research Group
Numerical Analysis
Copyright
© 2021 V.N.S.R. Dwarka, R.P.W.M. Tielen, M. Möller, Cornelis Vuik
DOI related publication
https://doi.org/10.1016/j.cma.2021.113694
More Info
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Publication Year
2021
Language
English
Copyright
© 2021 V.N.S.R. Dwarka, R.P.W.M. Tielen, M. Möller, Cornelis Vuik
Research Group
Numerical Analysis
Volume number
377
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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.