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 upon 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 realise full approximation storage (FAS) within the additive environment where, amortised, each grid vertex carrying degrees of freedom is read/written only once per iteration. The codes realise a single-touch policy. Among the features facilitated by matrix-free FAS is arbitrary dynamic mesh refinement (AMR) for all solver variants. AMR as enabler for full multigrid (FMG) cycling-the grid unfolds throughout the computation-allows us to reduce the cost per unknown per order of accuracy.The present paper primary contributes towards software realisation and design questions. Our experiments show that the consolidation of single-touch FAS, dynamic AMR and vectorisation-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.
This paper studies and analyzes a preconditioned Krylov solver for Helmholtz problems that are formulated with absorbing boundary layers based on complex coordinate stretching. The preconditioner problem is a Helmholtz problem where not only the coordinates in the absorbing layer have an imaginary part, but also the coordinates in the interior region. This results into a preconditioner problem that is invertible with a multigrid cycle. We give a numerical analysis based on the eigenvalues and evaluate the performance with several numerical experiments. The method is an alternative to the complex shifted Laplacian and it gives a comparable performance for the studied model problems.
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