Complex additive geometric multilevel solvers for Helmholtz equations on spacetrees

TL;DR

This paper presents a flexible, matrix-free, multilevel solver framework for Helmholtz equations that supports complex scaling, dynamic mesh refinement, and efficient computation, advancing software design for challenging PDE problems.

Contribution

It introduces a unified, matrix-free, additive multilevel solver implementation for Helmholtz equations with complex scaling and adaptive mesh refinement capabilities.

Findings

Efficient single-touch FAS scheme implemented with dynamic AMR.

Supports complex scaled Helmholtz operators with matrix-free methods.

Provides a formal, correct implementation blueprint for advanced solvers.

Abstract

We introduce a family of implementations of low order, additive, geometric multilevel solvers for systems of Helmholtz equations. Both grid spacing and arithmetics may comprise complex numbers and we thus can apply complex scaling techniques 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 pure additive multigrid over a hierarchical basis method into BPX with several multiscale smoothing variants within one code base. Pipelining allows us to realise a full approximation storage (FAS) scheme within the additive environment where, amortised, each grid vertex carrying degrees of freedom is read/written only once per iteration. The codes thus 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 for a non-trivial class of problems such as Helmholtz equations. Besides this validation, we put particular emphasis on a strict implementation formalism as well as some implementation correctness proofs.