An Efficient Multigrid Solver for Anisotropic Poisson Problems Using Plane Relaxation

Solving the Poisson equation is often the primary computational bottleneck in incompressible flow simulations. Although multigrid methods are highly efficient for elliptic equations, their performance deteriorates in the presence of anisotropy, which may arise from either the underlying physics or numerical discretization. In this study, we employ a plane relaxation method combined with different coarsening strategies to address varying degrees of anisotropy across the computational domain. Anisotropy is introduced by the discrete Laplace operator on non-uniform cylindrical grids. Results show that alternating plane relaxation with full coarsening offers greater robustness and achieves more consistent error reduction per cycle even on highly stretched grids than plane relaxation with semi-coarsening. We also present a parallel implementation of the method, focusing on optimizing the process layout at each multigrid level to minimize total time. As the number of grid points decreases on coarser levels, the number of active processes should be adjusted accordingly, which incurs communication overhead due to redistribution. A shortest-path algorithm is used to determine the process configuration that accounts for this cost at each level and selects the optimal layout.