Energy conserving Galerkin approximations for 2-D hydrodynamic and MHD Bénard convection

The upper bound on the minimum number of modes for a qualitatively correct Galerkin approximation to the two-dimensionl Bénard convection is discussed. For that case and the ext case, selection rules are introduced insuring that the approximation satisfies the same energy balance conditions as the exact solution. These rules together with the proof of boundedness of the Galerkin approximation help establish bounds on the amplitudes of the modes. Finally, results of some mathematical (numerical) experiments carried out in this context are discussed.