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

Dawson, Clint N.; Wheeler, Mary F.

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

Keywords:

two-grid methods; mixed finite element; error estimates; nonlinear parabolic equations; flow through porous media; implicit time discretizations