Monolithic Algebraic Multigrid Preconditioners for the Stokes Equations
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In this paper, we investigate a novel monolithic algebraic multigrid solver for the discrete Stokes problem discretized with stable mixed finite elements. The algorithm is based on the use of the low-order ℙ_1 iso1ptℙ_2/ ℙ_1 discretization as a preconditioner for a higher-order discretization, such as ℙ_2/ℙ_1. Smoothed aggregation algebraic multigrid is used to construct independent coarsenings of the velocity and pressure fields for the low-order discretization, resulting in a purely algebraic preconditioner for the high-order discretization (i.e., using no geometric information). Furthermore, we incorporate a novel block LU factorization technique for Vanka patches, which balances computational efficiency with lower storage requirements. The effectiveness of the new method is verified for the ℙ_2/ℙ_1 (Taylor-Hood) discretization in two and three dimensions on both structured and unstructured meshes. Similarly, the approach is shown to be effective when applied to the ℙ_2/ℙ_1^disc (Scott-Vogelius) discretization on 2D barycentrically refined meshes. This novel monolithic algebraic multigrid solver not only meets but frequently surpasses the performance of inexact Uzawa preconditioners, demonstrating the versatility and robust performance across a diverse spectrum of problem sets, even where inexact Uzawa preconditioners struggle to converge.
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