Premium
The influence of the stabilization parameter on the convergence factor of iterative methods for the solution of the discretized Stokes problem
Author(s) -
Vincent C.
Publication year - 1995
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.1650201104
Subject(s) - mathematics , discretization , preconditioner , eigenvalues and eigenvectors , iterated function , mathematical optimization , convergence (economics) , context (archaeology) , conjugate gradient method , operator (biology) , finite element method , upper and lower bounds , iterative method , mathematical analysis , paleontology , physics , quantum mechanics , economics , biology , economic growth , thermodynamics , biochemistry , chemistry , repressor , gene , transcription factor
This paper discusses the influence of the stabilization parameter on the convergence factor of various iterative methods for the solution of the Stokes problem discretized by the so‐called locally stabilized Q1‐P0 finite element. Our objective is to point out optimal parameters which ensure rapid convergence. The first part of the paper is concerned with the dual formulation of the problem. It gives the theoretical precision and practical developments of our stabilized context Uzawa‐type algorithm. We assert that the convergence factor of such a method is majored independently of the mesh size by a function of the stabilization parameter. Moreover, we point out that there exists an optimal value of this parameter that minimizes this upper bound. This gives a theoretical justification of pre‐existing numerical results. We show that the optimal parameter can be determined a priori. This is a key point when the method has to be implemented. Finally, we base an interpretation of the iterated penalty method numerical behaviour on some theoretical results about the minimum eigenvalue of the stabilized dual operator. This algorithm involves a penalty parameter and a stabilization parameter and we discuss a strategy for choosing optimal parameters. The mixed formulation of the problem is dealt with in the second part of the paper, which proposes several preconditioned conjugate‐gradient‐type methods. The indefinite character of the problem makes it intrinsically hard. However, if one chooses a suitable preconditioner, this difficulty is overcome, since the preconditioned operator becomes positive definite. We study the eigenvalue spectrum of the preconditioned operator and thereby the convergence factor of the algorithm. In contrast with the two previous formulations, we show that this convergence factor is majored independently of the stabilization parameter. More precisely, we point out convergence factors comparable with those obtained for Poisson‐type problems. Finally, we present a variant of the latter method which uses our so‐called macroblock‐type preconditioner. A comparison with the simple case of diagonal preconditioning is addressed and the improved performance of the macroblock‐type preconditioner is evidenced. Various 2D numerical experiments are given to corroborate the theories presented herein.