%0 Journal Article %1 verfurth1984combined %A Verfurth, Rudiger %D 1984 %J IMA Journal of Numerical Analysis %K 65f08-preconditioners-for-iterative-methods 65f10-iterative-methods-for-linear-systems 65n55-pdes-bvps-multigrid-methods-domain-decomposition 76d07-stokes-and-related-oseen-etc-flows %N 4 %P 441–455 %R 10.1093/imanum/4.4.441 %T A combined conjugate gradient - multi-grid algorithm for the numerical solution of the Stokes problem %U https://academic.oup.com/imajna/article-abstract/4/4/441/651488 %V 4 %X We consider an algorithm for the solution of a mixed finite-element approximation of the Stokes equations in a bounded, simply connected domain Ω ⊂ R2. The original indefinite problem for the velocity and pressure can be transformed into an equation Lp = g for the pressure involving a symmetric, positive definite, continuous linear operator L: L2(Ω)/R → L2(Ω)/R. We apply a conjugate gradient algorithm to this equation. Each evaluation of Lp requires the solution of two discrete Poisson equations. This is done approximately using a multigrid algorithm. The resulting iterative process has a convergence rate κ bounded away from one independently of the meshsize. Numerical experiments yield values for κ between 0-8 and 0-93. The generalization of the analysis to other mixed problems is obvious.

@article{verfurth1984combined, abstract = {We consider an algorithm for the solution of a mixed finite-element approximation of the Stokes equations in a bounded, simply connected domain Ω ⊂ R2. The original indefinite problem for the velocity and pressure can be transformed into an equation Lp = g for the pressure involving a symmetric, positive definite, continuous linear operator L: L2(Ω)/R → L2(Ω)/R. We apply a conjugate gradient algorithm to this equation. Each evaluation of Lp requires the solution of two discrete Poisson equations. This is done approximately using a multigrid algorithm. The resulting iterative process has a convergence rate κ bounded away from one independently of the meshsize. Numerical experiments yield values for κ between 0-8 and 0-93. The generalization of the analysis to other mixed problems is obvious.}, added-at = {2019-12-06T05:45:59.000+0100}, author = {Verfurth, Rudiger}, biburl = {https://www.bibsonomy.org/bibtex/24e1725862ecbe10682b92bcfc33a926f/gdmcbain}, doi = {10.1093/imanum/4.4.441}, interhash = {d271707c97a42a609e7a63639de385da}, intrahash = {4e1725862ecbe10682b92bcfc33a926f}, journal = {IMA Journal of Numerical Analysis}, keywords = {65f08-preconditioners-for-iterative-methods 65f10-iterative-methods-for-linear-systems 65n55-pdes-bvps-multigrid-methods-domain-decomposition 76d07-stokes-and-related-oseen-etc-flows}, month = oct, number = 4, pages = {441–455}, timestamp = {2019-12-06T05:46:13.000+0100}, title = { A combined conjugate gradient - multi-grid algorithm for the numerical solution of the Stokes problem}, url = {https://academic.oup.com/imajna/article-abstract/4/4/441/651488}, volume = 4, year = 1984 }