Multigrid solvers are among the most efficient methods for solving the Poisson equation, which is ubiquitous in computational physics. For example, in the context of incompressible flows, it is typically the costliest operation. The present document expounds upon the implementation of a flexible multigrid solver that is capable of handling any type of boundary conditions within murphy, a multiresolution framework for solving partial differential equations (PDEs) on collocated adaptive grids. The utilization of a Fourier-based direct solver facilitates the attainment of flexibility and enhanced performance by accommodating any combination of unbounded and semi-unbounded boundary conditions. The employment of high-order compact stencils contributes to the reduction of communication demands while concurrently enhancing the accuracy of the system. The resulting solver is validated against analytical solutions for periodic and unbounded domains. In conclusion, the solver has been demonstrated to demonstrate scalability to 16,384 cores within the context of leading European high-performance computing infrastructures.

翻译：多重网格求解器是求解泊松方程最高效的方法之一，该方程在计算物理学中普遍存在。例如，在不可压缩流动的背景下，它通常是计算代价最高的操作。本文阐述了一种灵活多重网格求解器的实现，该求解器能够在murphy（一种用于在并置自适应网格上求解偏微分方程的多分辨率框架）内处理任何类型的边界条件。通过采用基于傅里叶变换的直接求解器，能够适应无界和半无界边界条件的任意组合，从而实现了灵活性和性能提升。高阶紧致格式的使用有助于减少通信需求，同时提高系统精度。该求解器通过周期域和无界域的解析解进行了验证。总之，该求解器已在欧洲领先的高性能计算基础设施中展现出可扩展至16,384个计算核心的能力。