Error Estimates on a New Nonlinear Galerkin Method Based on Two-Grid Finite Elements

Abstract

A new nonlinear Galerkin method based on nite element discretization is presented in this paper for a class of second order nonlinear parabolic equations. The new scheme is based on two di erent nite element spaces de ned respectively on one coarse grid with grid size H and one ne grid with grid size h H. Nonlinearity and time dependence are both treated on the coarse space and only a xed stationary equation needs to be solved on the ne space at each time. With linear nite element discretizations, it is proved that the di erence between the new nonlinear Galerkin solution and the standard Galerkin solution in H 1 () norm is of the order of H 3 .

Postprocessing of a finite volume element method for semilinear parabolic problems

“…The postprocessing technique can be seen as a novel two-level or two-grid method, which involves an additional solution on a finer grid after the time evolution is finished. Unlike the traditional two-grid or two-level approaches, there is no communication from fine to coarse meshes until the end of time-marching [13,18,24]. This means that the extra cost of the postprocessing is relatively negligible when compared with the cost of computations from t = 0 to t = T on the coarser mesh.…”

Numerical analysis of a modified finite element nonlinear Galerkin method

“…The following properties, which are classical consequences of (3.2)-(3.4) (see [1,3,8,19]), will be very useful…”

“…where, as in [1,19], the intermediate space Xf! is the L 2 -orthogonal supplementary of X H in X h .…”

“…These questions have mainly been addressed in the case of spectral Fourier discretizations( see [4][5][6]15,17,20] and the references therein) . The case of finite element approximations for general nonlinear evolution equations are considered in [18][19]. The case of mixed finite elements approximations for the Navier-Stokes equations is treated in [1,[10][11][12].…”

Adaptive multilevel methods in space and time for parabolic problems-the periodic case

“…Such a discretization allows one to simply define a two-level spatial decomposition; but, the authors believe that the general idea of the multilevel method extends to other types of spatial discretization and, in particular, to finite element methods (see Marion and Xu [11] for a first step in that direction, based on L 2 orthogonal decomposition).…”