Distributive smoothers in multigrid for problems with dominating grad–div operators

In this paper, we present efficient multigrid methods for systems of partial differential equations that are governed by a dominating grad–div operator. In particular, we show that distributive smoothing methods give multigrid convergence factors that are independent of problem parameters and of the mesh sizes in space and time. The applications range from model problems to secondary consolidation Biot's model. We focus on the smoothing issue and mainly solve academic problems on Cartesian-staggered grids. Copyright © 2008 John Wiley & Sons, Ltd.

[1]  Joachim Schöberl,et al.  Multigrid methods for a parameter dependent problem in primal variables , 1999, Numerische Mathematik.

[2]  Petr N. Vabishchevich,et al.  A stabilized method for a secondary consolidation Biot's model , 2008 .

[3]  R. Showalter.  Diffusion in Poro-Elastic Media , 2000 .

[4]  E. F. Kaasschieter,et al.  Squeezing a Sponge: A Three-Dimensional Solution in Poroelasticity , 2003 .

[5]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[6]  Abimael F. D. Loula,et al.  Improved accuracy in finite element analysis of Biot's consolidation problem , 1992 .

[7]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[8]  S. Vanka.  Block-implicit multigrid solution of Navier-Stokes equations in primitive variables , 1986 .

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

[10]  Cornelis W. Oosterlee,et al.  A stabilized difference scheme for deformable porous media and its numerical resolution by multigrid methods , 2008 .

[11]  Abimael F. D. Loula,et al.  On stability and convergence of finite element approximations of biot's consolidation problem , 1994 .

[12]  Ralf Hiptmair,et al.  Multilevel methods for mixed finite elements in three dimensions , 1999, Numerische Mathematik.

[13]  R. Lazarov,et al.  On convergence of certain finite volume difference discretizations for 1D poroelasticity interface problems , 2007 .

[14]  Cornelis W. Oosterlee,et al.  An Efficient Multigrid Solver based on Distributive Smoothing for Poroelasticity Equations , 2004, Computing.

[15]  Hélène Barucq,et al.  On nonlinear Biot's consolidation models , 2005 .

[16]  Gabriel Wittum,et al.  On the convergence of multi-grid methods with transforming smoothers , 1990 .

[17]  P. Vassilevski,et al.  Multilevel iterative methods for mixed finite element discretizations of elliptic problems , 1992 .

[18]  K. Terzaghi.  Theoretical Soil Mechanics , 1943 .

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

[20]  M. Shashkov,et al.  Adjoint operators for the natural discretizations of the divergence gradient and curl on logically rectangular grids , 1997 .

[21]  Ping Lin.  A Sequential Regularization Method for Time-Dependent Incompressible Navier--Stokes Equations , 1997 .

[22]  R. S. Falk,et al.  PRECONDITIONING IN H (div) AND APPLICATIONS , 1997 .

[23]  Maxim A. Olshanskii,et al.  Grad–div stabilization and subgrid pressure models for the incompressible Navier–Stokes equations , 2009 .

[24]  John H. Cushman,et al.  Multiscale flow and deformation in hydrophilic swelling porous media , 1996 .

[25]  Mikhail Shashkov,et al.  The effect of inner products for discrete vector fields on the accuracy of mimetic finite difference methods , 2001 .

[26]  Tongchun Li,et al.  A new stabilized enhanced strain element with equal order of interpolation for soil consolidation problems , 2003 .

[27]  F. Gaspar,et al.  A finite difference analysis of Biot's consolidation model , 2003 .

[28]  Cornelis W. Oosterlee,et al.  An efficient multigrid solver for a reformulated version of the poroelasticity system , 2007 .

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

[30]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[31]  Cornelis W. Oosterlee,et al.  A systematic comparison of coupled and distributive smoothing in multigrid for the poroelasticity system , 2004, Numer. Linear Algebra Appl..

[32]  Stephen F. McCormick,et al.  Multilevel adaptive methods for partial differential equations , 1989, Frontiers in applied mathematics.

[33]  M. Biot.  General Theory of Three‐Dimensional Consolidation , 1941 .

[34]  E. F. Kaasschieter,et al.  Squeezing a sponge : a three-dimensional analytic solution in poroelasticity , 2000 .

[35]  R. Showalter,et al.  DIFFUSION IN DEFORMING POROUS MEDIA , 2002 .

[36]  Douglas N. Arnold,et al.  Multigrid in H (div) and H (curl) , 2000, Numerische Mathematik.

[37]  B. Schrefler,et al.  The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media , 1998 .

[38]  Vidar Thomée,et al.  Asymptotic behavior of semidiscrete finite-element approximations of Biot's consolidation problem , 1996 .

[39]  Olaf Kolditz,et al.  Finite element analysis of poro-elastic consolidation in porous media: Standard and mixed approaches , 2006 .

[40]  Maxim A. Olshanskii,et al.  Grad-div stablilization for Stokes equations , 2003, Math. Comput..

[41]  Christian Wieners,et al.  Robust Multigrid Methods for Nearly Incompressible Elasticity , 2000, Computing.

[42]  F. Gaspar,et al.  Staggered grid discretizations for the quasi-static Biot's consolidation problem , 2006 .