Abstract
We review a number of preconditioners for the advection-diffusion operator and for the Schur complement matrix, which, in turn, constitute the building blocks for Constraint and Triangular Preconditioners to accelerate the iterative solution of the discretized and linearized Navier-Stokes equations. An intensive numerical testing is performed onto the driven cavity problem with low values of the viscosity coefficient. We devise an efficient multigrid preconditioner for the advection-diffusion matrix, which, combined with the commuted BFBt Schur complement approximation, and inserted in a 2×2 block preconditioner, provides convergence of the Generalized Minimal Residual (GMRES) method in a number of iteration independent of the meshsize for the lowest values of the viscosity parameter. The low-rank acceleration of such preconditioner is also investigated, showing its great potential.
Highlights
The task of numerically solving the Navier–Stokes equations is of fundamental importance in many scientific and industrial applications; the strong nonlinearity of the equations and the lack of any theoretical result about existence and regularity of the solutions leaves the scene only to numerical approximations
We focus in finding a scalable preconditioner for these saddle point linear systems: many preconditioners have already been developed and tested successfully, for instance in [1,2,3,4,5]; we choose to use the Constraint Preconditioner, already analyzed in all its forms in [6,7,8], and the more popular, in the Fluid Dynamics community, Block Triangular Preconditioner, employed e.g., in [9,10]
The results developed for ICP and Block Triangular Preconditioner (BTP) preconditioned matrices show that unit eigenvalues are perturbed by the presence of error matrices EF and ES whose norms, in the case of scalable preconditioners for the blocks, swill be small and independent of the mesh discretization parameter h
Summary
The task of numerically solving the Navier–Stokes equations is of fundamental importance in many scientific and industrial applications; the strong nonlinearity of the equations and the lack of any theoretical result about existence and regularity of the solutions leaves the scene only to numerical approximations. One of the approaches to the numerical solution of the Navier–Stokes equations is given by the Finite Element Method, which, after a linearization of the nonlinear terms, gives rise to a saddle point linear system: this particular system has a structure that appears in many other problems, but in this context it is possible to exploit the underlying continuous formulation to develop efficient preconditioners in the framework of iterative solvers. The generalization of Multigrid preconditioners to these kind of situations requires a robust smoother, which can be built using a stationary method involving a pattern that follows the convective flow This approach has already been tested in [2,12,13]. We will use the symbol M to denote a preconditioner which approximates a given coefficient matrix A( M ≈ A), while the symbol P will refer to the preconditioner in its inverse form (P ≈ A−1 )
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