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**Two-grid methods for mixed finite element approximations of nonlinear parabolic equations.**
*(English)*
Zbl 0817.65080

Keyes, David E. (ed.) et al., Domain decomposition methods in scientific and engineering computing. Proceedings of the 7th international conference on domain decomposition, October 27-30, 1993, Pennsylvania State University, PA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 180, 191-203 (1994).

Summary: Mixed finite element approximation of nonlinear parabolic equations is discussed. The equation considered is a prototype of a model that arises in flow through porous media. A two-grid approximation scheme is developed and analyzed for implicit time discretizations. In this approach, the full nonlinear system is solved on a “coarse” grid of size \(H\). The nonlinearities are expanded about the coarse grid solution, and the resulting linear but nonsymmetric system is solved on a “fine” grid of size \(h\).

Error estimates are derived which demonstrate that the error is \({\mathcal O}(h^{k+ 1}+ H^{2(k+ 1)- d/2}+ \Delta t)\), where \(k\) is the degree of the approximating space for the primary variable and \(d\) is spatial dimension, with \(k\geq 1\) for \(d\geq 2\). For the \(RT_ 0\) space \((k= 0)\) on rectangular domains, we present a modified scheme for treating the coarse grid problem. Here we show that the error is \({\mathcal O}(h+ H^{3- d/2}+ \Delta t)\). The above estimates are useful for determining an appropriate \(H\) for the coarse grid problem.

For the entire collection see [Zbl 0809.00026].

Error estimates are derived which demonstrate that the error is \({\mathcal O}(h^{k+ 1}+ H^{2(k+ 1)- d/2}+ \Delta t)\), where \(k\) is the degree of the approximating space for the primary variable and \(d\) is spatial dimension, with \(k\geq 1\) for \(d\geq 2\). For the \(RT_ 0\) space \((k= 0)\) on rectangular domains, we present a modified scheme for treating the coarse grid problem. Here we show that the error is \({\mathcal O}(h+ H^{3- d/2}+ \Delta t)\). The above estimates are useful for determining an appropriate \(H\) for the coarse grid problem.

For the entire collection see [Zbl 0809.00026].

### MSC:

65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

76M10 | Finite element methods applied to problems in fluid mechanics |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

35K55 | Nonlinear parabolic equations |

76S05 | Flows in porous media; filtration; seepage |