Novel mass-based multigrid relaxation schemes for the Stokes equations

In this work, we propose three novel block-structured multigrid relaxation schemes based on distributive relaxation, Braess-Sarazin relaxation, and Uzawa relaxation, for solving the Stokes equations discretized by the mark-and-cell scheme. In our earlier work [11], we discussed these three types of relaxation schemes, where the weighted Jacobi iteration is used for inventing the Laplacian involved in the Stokes equations. In [11], we show that the optimal smoothing factor is 3 5 for distributive weighted-Jacobi relaxation and inexact Braess-Sarazin relaxation, and is

[1]  Joachim Schöberl,et al.  On Schwarz-type Smoothers for Saddle Point Problems , 2003, Numerische Mathematik.

[2]  Panayot S. Vassilevski,et al.  A new approach for solving stokes systems arising from a distributive relaxation method , 2011 .

[3]  S. MacLachlan,et al.  Monolithic multigrid for a reduced-quadrature discretization of poroelasticity , 2021, SIAM J. Sci. Comput..

[4]  A. Brandt,et al.  Multigrid Solutions to Elliptic Flow Problems , 1979 .

[5]  Cornelis W. Oosterlee,et al.  On an Uzawa smoother in multigrid for poroelasticity equations , 2017, Numer. Linear Algebra Appl..

[6]  Chen Greif,et al.  A closed‐form multigrid smoothing factor for an additive Vanka‐type smoother applied to the Poisson equation , 2021, Numerical Linear Algebra with Applications.

[7]  Walter Zulehner,et al.  A Class of Smoothers for Saddle Point Problems , 2000, Computing.

[8]  Long Chen.  INTRODUCTION TO MULTIGRID METHODS , 2005 .

[9]  Ming Wang,et al.  A Multigrid Solver based on Distributive Smoother and Residual Overweighting for Oseen Problems , 2015 .

[10]  Ming Wang,et al.  Multigrid Methods for the Stokes Equations using Distributive Gauss–Seidel Relaxations based on the Least Squares Commutator , 2013, Journal of Scientific Computing.

[11]  G. Wittum.  Multi-grid methods for stokes and navier-stokes equations , 1989 .

[12]  Yvan Notay,et al.  A Simple and Efficient Segregated Smoother for the Discrete Stokes Equations , 2014, SIAM J. Sci. Comput..

[13]  G. Golub,et al.  Inexact and preconditioned Uzawa algorithms for saddle point problems , 1994 .

[14]  Yunhui He,et al.  Low‐order preconditioning of the Stokes equations , 2021, Numerical Linear Algebra with Applications.

[15]  F. Musy,et al.  A Fast Solver for the Stokes Equations Using Multigrid with a UZAWA Smoother , 1985 .

[16]  Scott P. MacLachlan,et al.  Monolithic Multigrid Methods for Two-Dimensional Resistive Magnetohydrodynamics , 2016, SIAM J. Sci. Comput..

[17]  Apostol T. Vassilev,et al.  Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems , 1997 .

[18]  Scott P. MacLachlan,et al.  Local Fourier analysis for mixed finite-element methods for the Stokes equations , 2019, J. Comput. Appl. Math..

[19]  Scott P. MacLachlan,et al.  Local Fourier analysis of block‐structured multigrid relaxation schemes for the Stokes equations , 2018, Numer. Linear Algebra Appl..

[20]  D. Braess,et al.  An efficient smoother for the Stokes problem , 1997 .

[21]  Cornelis W. Oosterlee,et al.  Multigrid Methods for the Stokes System , 2006, Computing in Science & Engineering.