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Complex additive geometric multilevel solvers for Helmholtz equations on spacetrees

Reps, Bram; Weinzierl, Tobias

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Bram Reps


We introduce a family of implementations of low-order, additive, geometric multilevel solvers for systems of Helmholtz equations arising from Schrödinger equations. Both grid spacing and arithmetics may comprise complex numbers, and we thus can apply complex scaling to the indefinite Helmholtz operator. Our implementations are based on the notion of a spacetree and work exclusively with a finite number of precomputed local element matrices. They are globally matrix-free. Combining various relaxation factors with two grid transfer operators allows us to switch from additive multigrid over a hierarchical basis method into a Bramble-Pasciak-Xu (BPX)-type solver, with several multiscale smoothing variants within one code base. Pipelining allows us to realize full approximation storage (FAS) within the additive environment where, amortized, each grid vertex carrying degrees of freedom is read/written only once per iteration. The codes realize a single-touch policy. Among the features facilitated by matrix-free FAS is arbitrary dynamic mesh refinement (AMR) for all solver variants. AMR as an enabler for full multigrid (FMG) cycling—the grid unfolds throughout the computation—allows us to reduce the cost per unknown. The present work primary contributes toward software realization and design questions. Our experiments show that the consolidation of single-touch FAS, dynamic AMR, and vectorization-friendly, complex scaled, matrix-free FMG cycles delivers a mature implementation blueprint for solvers of Helmholtz equations in general. For this blueprint, we put particular emphasis on a strict implementation formalism as well as some implementation correctness proofs.


Reps, B., & Weinzierl, T. (2017). Complex additive geometric multilevel solvers for Helmholtz equations on spacetrees. ACM Transactions on Mathematical Software, 44(1), Article 2.

Journal Article Type Article
Acceptance Date Aug 19, 2016
Online Publication Date Mar 27, 2017
Publication Date Mar 27, 2017
Deposit Date Aug 19, 2016
Publicly Available Date Sep 20, 2016
Journal ACM Transactions on Mathematical Software
Print ISSN 0098-3500
Electronic ISSN 1557-7295
Publisher Association for Computing Machinery (ACM)
Peer Reviewed Peer Reviewed
Volume 44
Issue 1
Article Number 2
Related Public URLs


Accepted Journal Article (4 Mb)

Copyright Statement
© ACM, 2017. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in ACM Transactions on Mathematical Software, Volume 44 Issue 1, March 2017,
Article No. 2

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