## 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].

### 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