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

Dwarka, Vandana; Tielen, Roel; Möller, Matthias; Vuik, C.

doi:10.1016/j.cma.2021.113694

Cited by 6 publications

(15 citation statements)

References 41 publications

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“…where A is the CSLP (41) with β = 1/(3k) = 7.95 × 10 −5 , as suggested in [12]. After only 1 outer iteration of GMRES the relative residual of preconditioned system (42) was 3.729 × 10 −10 .…”

Section: Resultsmentioning

confidence: 99%

“…In order to obtain a good preconditioner, β/k needs to be sufficiently small [17]. In [11] experiments are done with β = √ k. Moreover, to obtain wavenumber independent convergence, the complex shift β used in [12] is of order O(k) −1 .…”

Section: Galerkin Methods With B-spline Functionsmentioning

confidence: 99%

“…The literature dealing with different aspects of the solution of Helmholtz equation with IgA is very recent [24], [8], [10], [11] and [12]. In [24] the performance of IgA to solve exterior scattering problems is investigated, using an absorbing boundary condition on a fictitious boundary to truncate the infinite space.…”

Section: Related Workmentioning

confidence: 99%

“…Moreover, the pollution error is significantly reduced with IgA, even when it is not completely eliminated. In a very recent paper [12], the focus of the research is on the numerical solution of the linear system derived from IgA discretization of Helmholtz equation. The system is solved with GMRES and its convergency is accelerated using a deflation technique, combined with the approximated computation of the inverse of the CSLP with a geometric multigrid method.…”

Section: Related Workmentioning

confidence: 99%

“…Hence Fredholm theory can be applied, to show that a solution to the variational problem (12) exists since the homogeneous adjoint problem has an unique solution. Details of the proof can be found in [21].…”

Section: Variational Formulationmentioning

confidence: 99%

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Isogeometric simulation of acoustic radiation

Mederos¹,

Hernández²,

Sarlabous³

et al. 2021

Preprint

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In this paper we discuss the numerical solution on a simple 2D domain of the Helmoltz equation with mixed boundary conditions. The so called radiation problem depends on the wavenumber constant parameter k and it is inspired here by medical applications, where a transducer emits a pulse at a given frequency. This problem has been successfully solved in the past with the classical Finite Element Method (FEM) for relative small values of k. But in modern applications the values of k can be of order of thousands and FEM faces up several numerical difficulties. To overcome these difficulties we solve the radiation problem using the Isogeometric Analysis (IgA), a kind of generalization of FEM. Starting with the variational formulation of the radiation problem, we show with details how to apply the isogeometric approach in order to compute the coefficients of the approximated solution of radiation problem in terms of the B-spline basis functions. Our implementation of IgA using GeoPDEs software shows that isogeometric approach is superior than FEM, since it is able to reduce significatively the pollution error, especially for high values of k, producing additionally smoother solutions which depend on less degrees of freedom.

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Section: Resultsmentioning

confidence: 99%