Error estimates on a new nonlinear Galerkin method based on two-grid finite elements

A new nonlinear Galerkin method based on finite element discretization is presented in this paper for semilinear parabolic equations. The new scheme is based on two different finite element spaces defined respectively on one coarse grid with grid size H and one fine grid with grid size $h \ll H$. Nonlinearity and time dependence are both treated on the coarse space and only a fixed stationary equation needs to be solved on the fine space at each time. With linear finite element discretizations, it is proved that the difference between the new nonlinear Galerkin solution and the standard Galerkin solution in $H^1 (\Omega )$ norm is of the order of $H^3 $.

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