This work presents a multigrid preconditioned high order immersed finite difference solver to accurately and efficiently solve the Poisson equation on complex 2D and 3D domains. The solver employs a low order Shortley-Weller multigrid method to precondition a high order matrix-free Krylov subspace solver. The matrix-free approach enables full compatibility with high order IIM discretizations of boundary and interface conditions, as well as high order wavelet-adapted multiresolution grids. Through verification and analysis on 2D domains, we demonstrate the ability of the algorithm to provide high order accurate results to Laplace and Poisson problems with Dirichlet, Neumann, and/or interface jump boundary conditions, all effectively preconditioned using the multigrid method. We further show that the proposed method is able to efficiently solve high order discretizations of Laplace and Poisson problems on complex 3D domains using thousands of compute cores and on multiresolution grids. To our knowledge, this work presents the largest problem sizes tackled with high order immersed methods applied to elliptic partial differential equations, and the first high order results on 3D multiresolution adaptive grids. Together, this work paves the way for employing high order immersed methods to a variety of 3D partial differential equations with boundary or inter-face conditions, including linear and non-linear elasticity problems, the incompressible Navier-Stokes equations, and fluid-structure interactions.

翻译：本研究提出了一种多重网格预条件高阶浸入有限差分求解器，用于在复杂二维与三维域上精确高效地求解泊松方程。该求解器采用低阶Shortley-Weller多重网格方法对高阶无矩阵Krylov子空间求解器进行预条件处理。无矩阵方法使其能够完全兼容边界与界面条件的高阶IIM离散格式，以及高阶小波自适应多分辨率网格。通过对二维域的验证与分析，我们证明了该算法能够为带有Dirichlet、Neumann和/或界面跳跃边界条件的Laplace与泊松问题提供高阶精确解，且全部通过多重网格方法实现有效预条件处理。我们进一步表明，所提方法能够利用数千个计算核心在复杂三维域及多分辨率网格上高效求解Laplace与泊松问题的高阶离散格式。据我们所知，本研究呈现了当前采用高阶浸入方法处理椭圆型偏微分方程的最大规模问题，并首次在三维多分辨率自适应网格上获得高阶结果。综上所述，本工作为将高阶浸入方法应用于各类带边界或界面条件的三维偏微分方程（包括线性和非线性弹性问题、不可压缩Navier-Stokes方程以及流固耦合问题）开辟了道路。