Fault-Tolerant Parallel Multigrid Method on Unstructured Adaptive Mesh

As the generation of exascale high-performance clusters begins, it has become evident that numerical algorithms will greatly benefit from built-in resilience features that can handle system faults. Prior studies of fault-tolerant multigrid methods have focused on structured grids. In this work, however, we study the resilience of multigrid solvers on unstructured grids with adaptive refinement. The challenge lies in the fact that unstructured grids distributed across multiple processors may manifest as local hierarchical grids with unaligned boundaries. Our numerical experiments highlight that this disparity can result in divergence when employing standard local multigrid for fault recovery. We analyze this phenomenon by using an energy control condition. To tackle the divergence issue, we propose a simple variation of the multigrid V-cycle that scales the coarse problem. We present a convergence proof for the new algorithm. By implementing this new method for local recovery, our numerical experiments confirm that convergence can be recovered on unstructured grids while the algorithm agrees with the standard multigrid V-cycle on grids with aligned boundaries. More importantly, the impact of a fault can be mitigated and delays in the global multigrid iterations can be reduced. Finally, we investigate how local regions within the adaptive mesh, associated with different faulty processors, affect the effectiveness of fault recovery.