Iterative methods for the solution of Stokes equations. (English) Zbl 0687.76026
Summary: The purpose of this paper is to present preconditioned conjugate gradient algorithms for the numerical solution of mixed finite element approximation of the Stokes problem considered as a saddle-point problem. Arrow-Hurwicz’s algorithm is an efficient method for finding a saddle- point; its convergence can be proven with a proper choice of parameters. However, it is not very popular for numerical codes because those parameters are not found naturally. On the other hand, the conjugate gradient method has several very enjoyable features when regarded as an iterative method for the solution of discretized 3-D partial differential equations. But its convergence requires the positivity of the system, which is not the case with the Stokes problem. We can, however, use the preconditioned conjugate gradient method with proper preconditioning even in the non-positive case, thus obtaining what can be seen as a variant of Arrow-Hurwicz’s method.
MSC:
| 76D07 | Stokes and related (Oseen, etc.) flows |
| 35Q99 | Partial differential equations of mathematical physics and other areas of application |
| 76M99 | Basic methods in fluid mechanics |
Keywords:
preconditioned conjugate gradient algorithms; numerical solution of mixed finite element approximation; Stokes problem; saddle-point problem; Arrow-Hurwicz’s algorithmReferences:
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