Implementation of arbitrary inner product in the global Galerkin method for incompressible Navier-Stokes equations

The global Galerkin or weighted residuals method applied to the incompressible Navier-Stokes equations is considered. The basis functions are assumed to be divergence-free and satisfy all the boundary conditions. The method is formulated for an arbitrary inner product, so that the pressure cannot be eliminated by Galerkin projections on a divergence-free basis. A proposed straightforward procedure for the elimination of the pressure reduces the problem to an ODE system without algebraic constraints. To illustrate the applicability and the robustness of the numerical approach and to show that numerical solutions with unit and non-unit weight functions yield similar results the driving lid cavity and natural convection benchmark problems are solved using the unit and Chebyshev weight functions. Further implications of the proposed Galerkin formulation are discussed.