# A parallel shared-memory implementation of a high-order accurate solution technique for variable coefficient Helmholtz problems

## TL;DR

This paper introduces a parallel shared-memory implementation of the Hierarchical Poincaré-Steklov (HPS) method, enabling high-accuracy solutions to variable coefficient Helmholtz problems on large-scale systems.

## Contribution

It presents the first parallel shared-memory implementation of the HPS method, facilitating large-scale Helmholtz problem simulations with high accuracy.

## Key findings

- Achieved efficient parallel performance on desktop hardware.
- Demonstrated scalability for large Helmholtz problems.
- Validated high-order accuracy without pollution effect.

## Abstract

The recently developed Hierarchical Poincar\'e-Steklov (HPS) method is a high-order discretization technique that comes with a direct solver. Results from previous papers demonstrate the method's ability to solve Helmholtz problems to high accuracy without the so-called pollution effect. While the asymptotic scaling of the direct solver's computational cost is the same as the nested dissection method, serial implementations of the solution technique are not practical for large scale numerical simulations. This manuscript presents the first parallel implementation of the HPS method. Specifically, we introduce an approach for a shared memory implementation of the solution technique utilizing parallel linear algebra. This approach is the foundation for future large scale simulations on supercomputers and clusters with large memory nodes. Performance results on a desktop computer (resembling a large memory node) are presented.