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Generalized Optimal AMG Convergence Theory for Stokes Equations Using Smooth Aggregation and Vanka Relaxation Strategies by Ali, Ahsan; Brannick, James J.; Kahl, Karsten; Krzysik, Oliver A.; Schroder, Jacob B.; Southworth, Ben S.; Voronin, Alexey is a scholarly article available to read on EtoBox.

What is Generalized Optimal AMG Convergence Theory for Stokes Equations Using Smooth Aggregation and Vanka Relaxation Strategies about?

This paper discusses our recent generalized optimal algebraic multigrid (AMG) convergence theory applied to the steady-state Stokes equations discretized using Taylor-Hood elements ($\pmb{ \mathbb{P}}_2/\mathbb{P}_{1}$). The generalized theory is founded on matrix-induced orthogonality of the left and right eigenvectors of a generalized eigenvalue problem involving the system matrix and relaxation operator. This framework establishes a rigorous lower bound on the spectral radius of the two-grid error-propagation operator, enabling precise predictions of the convergence rate for symmetric indefinite problems, such as those arising from saddle-point systems. We apply this theory to the recently developed monolithic smooth aggregation AMG (SA-AMG) solver for Stokes, constructed using evolution-based strength of connection, standard aggregation, and smoothed prolongation. The performance of these solvers is evaluated using additive and multiplicative Vanka relaxation strategies. Additive Vanka relaxation constructs patches algebraically on each level, resulting in a nonsymmetric relaxation operator due to the partition of unity being applied on one side of the block-diagonal matrix. Al

Author

Ali, Ahsan; Brannick, James J.; Kahl, Karsten; Krzysik, Oliver A.; Schroder, Jacob B.; Southworth, Ben S.; Voronin, Alexey

Published

2025

Language

EN