The problem of feedback controlled Rayleigh–Bénard convection is considered. For this problem with the simple flow structure in the vertical direction, a Galerkin method that uses only a few basis functions in this direction is presented. This approximation yields considerable simplification of the problem, explicitly incorporates the non-classical boundary conditions at the horizontal boundaries of the fluid layer resulting from feedback control and reduces the dimension of the original problem by one. This method is in spirit very similar to lubrication theory, where the simple laminar flow in the vertical direction is integrated out across the height of the fluid layer.

Using a minimal set of appropriate basis functions to capture the nonlinear behaviour of the flow, we investigate the effects of feedback control on amplitude, wavelength and selection of patterns via weakly nonlinear analysis and numerical simulations of the resulting dimension-reduced problems in two and three dimensions.

In the second part of this study we discuss the derivation of the appropriate basis functions and prove convergence of the Galerkin scheme.