Monolithic Multigrid Preconditioners for High-Order Discretizations of Stokes Equations
<|reference_start|>Monolithic Multigrid Preconditioners for High-Order Discretizations of Stokes Equations: This work introduces and assesses the efficiency of a monolithic $ph$MG multigrid framework designed for high-order discretizations of stationary Stokes systems using Taylor-Hood and Scott-Vogelius elements. The proposed approach integrates coarsening in both approximation order ($p$) and mesh resolution ($h$), to address the computational and memory efficiency challenges that are often encountered in conventional high-order numerical simulations. Our numerical results reveal that $ph$MG offers significant improvements over traditional spatial-coarsening-only multigrid ($h$MG) techniques for problems discretized with Taylor-Hood elements across a variety of problem sizes and discretization orders. In particular, the $ph$MG method exhibits superior performance in reducing setup and solve times, particularly when dealing with higher discretization orders and unstructured problem domains. For Scott-Vogelius discretizations, while monolithic $ph$MG delivers low iteration counts and competitive solve phase timings, it exhibits a discernibly slower setup phase when compared to a multilevel (non-monolithic) full-block-factorization (FBF) preconditioner where $ph$MG is employed only for the velocity unknowns. This is primarily due to the setup costs of the larger mixed-field relaxation patches with monolithic $ph$MG versus the patch setup costs with a single unknown type for FBF.<|reference_end|>
@article{voronin2024monolithic,
    title={Monolithic Multigrid Preconditioners for High-Order Discretizations of
  Stokes Equations},
    author={Alexey Voronin, Graham Harper, Scott MacLachlan, Luke N. Olson,
  Raymond S. Tuminaro},
    journal={arXiv preprint arXiv:2407.07253},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.07253},
    primaryClass={math.NA cs.NA}
}