A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem

Abstract

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractThe recently introduced divergence-conforming B-spline discretizations allow the construction of smooth discrete velocity-pressure pairs for viscous incompressible flows that are at the same time inf − sup stable and pointwise divergence-free. When applied to the discretized Stokes problem, these spaces generate a symmetric and indefinite saddle-point linear system. The iterative method of choice to solve such system is the Generalized Minimum Residual Method. This method lacks robustness, and one remedy is to use preconditioners. For linear systems of saddle-point type, a large family of preconditioners can be obtained by using a block factorization of the system. In this paper, we show how the nesting of "black-box" solvers and preconditioners can be put together in a block triangular strategy to build a scalable block preconditioner for the Stokes system discretized by divergence-conforming B-splines. Besides the known cavity flow problem, we used for benchmark flows defined on complex geometries: an eccentric annulus and hollow torus of an eccentric annular cross-section.

A comparison of block preconditioners for isogeometric analysis discretizations of the incompressible Navier–Stokes equations

“…The question we try to answer in this paper is whether they are applicable to the IgA discretizations of high degree and high order of continuity with the same success. A similar study was done for IgA discretizations of the Stokes equations by Côrtes et al, 16 where the authors combine a block triangular strategy with several “black‐box” solvers to get a scalable preconditioner. We consider the steady and unsteady Navier–Stokes equations linearized by Picard method and present a comparison of ideal versions of several block preconditioners.…”

Non-body-fitted fluid–structure interaction: Divergence-conforming B-splines, fully-implicit dynamics, and variational formulation

“…Our parallel implementation is built on top of PetIGA [71] and PetIGA-MF [72,73,74], which gives us access to the preconditioners of the scientific library PETSc [75] and the ones in other libraries as the BoomerAMG package of hypre [76] and the ML library [77]. To linearize the residual R K , we use a Newton-Raphson method.…”

“…As linear solver for the kinematic equation, we use the GMRES method [79] with an incomplete LU preconditioner. As linear solver for the momemtum and mass conservation equations, we use the scalable block-preconditioning strategy defined in [73] for divergence-conforming B-splines.…”

“…(81) is solved. As in [73], the iterative method used to solve the linear system is stopped when the L 2 norm of the unpreconditioned residual is less than a certain value a tol , which is set to 10 −10 in all our simulations unless when mentioned otherwise. e V C inherits the error coming from e DIV (which in our method is negligible) and adds the discretization error of solving Eq.…”

A geometric multigrid method for isogeometric compatible discretizations of the generalized Stokes and Oseen problems

“…In fact, only the performance of Krylov subspace methods in conjunction with block preconditioners has been investigated in prior work. 15,16 The objective of the current work is to introduce an optimally efficient linear solution procedure for isogeometric compatible discretizations of the generalized Stokes and Oseen problems. It should be noted that there are many different candidates in this regard.…”