An ILU preconditioner with coupled node fill‐in for iterative solution of the mixed finite element formulation of the 2D and 3D Navier‐Stokes equations

Abstract

SUMMARYIn the present paper, preconditioning of iterative equation solvers for the Navier-Stokes equations is investigated. The Navier-Stokes equations are solved for the mixed finite element formulation. The linear equation solvers used are the orthomin and the Bi-CGSTAB algorithms. The storage structure of the equation matrix is given special attention in order to avoid swapping and thereby increase the speed of the preconditioner. The preconditioners considered are Jacobian, SSOR and incomplete LU preconditioning of the matrix associated with the velocities. A new incomplete LU preconditioning with fill-in for the pressure matrix at locations in the matrix where the corner nodes are coupled is designed. For all preconditioners, inner iterations are investigated for possible improvement of the preconditioning. Numerical experiments are executed both in two and three dimensions.

Adaptive Linearization and Grid Iterations With the Tri-Tree Multigrid Refinement-Recoarsement Algorithm for the Navier-Stokes Equations

“…7 The present work shows that the computer time used by the equation solver is proportional to the number of unknowns and that the degree of non-linearity plays a minor role as long as the element Reynolds number is below a reasonable limit, less than 10.…”

“…6 The purpose of the present work is eventually to develop algorithms suited for multiprocessing. In previous papers a new tri-tree grid generator algorithm 6 and a new incomplete preconditioning algorithm 7 have been described. The tri-tree grid generation algorithm was well suited to organizing and structuring grids at different levels of refinement.…”

A comparison of preconditioners for incompressible Navier–Stokes solvers

“…To avoid this problem, it is better to use a suitable a priori reordering of unknowns. As pointed out by Wille and others [13][14][15], pivoting is not necessary when the unknowns are ordered in the sequence, so that all velocity unknowns come first and then all the pressure unknowns like in the block preconditioners. The reason being that during (incomplete) factorization the zeros at the main diagonal will vanish, provided fill-in is allowed based on the connectivity of nodal points rather than actual zeros in the matrix.…”

“…The same may happen in the case of ILU factorization. Wille and coworkers [13][14][15] use a node-renumbering technique in combination with a reordering of unknowns that is common for block preconditioners. This method can be applied to both direct and ILU-preconditioned iterative methods.…”

SIMPLE‐type preconditioners for the Oseen problem

“…In general algebraic preconditioners are based on ILU factorization of the coefficient matrix. In order to avoid problems with zeros on the main diagonal, either dynamic pivoting or a clever apriori reordering technique has to be applied [1][2][3][4][5][6][7][8][9]. In [9] we published an a-priori reordering technique (SILU), that converges fast for small to mid-sized grids.…”