Conservation properties for the Galerkin and stabilised forms of the advection-diffusion and incompressible Navier-Stokes equations

A common criticism of continuous Galerkin finite element methods is their perceived lack of conservation. This may in fact be true for incompressible flows when advective, rather than conservative, weak forms are employed. However, advective forms are often preferred on grounds of accuracy despite violation of conservation. It is shown here that this deficiency can be easily remedied, and conservative procedures for advective forms can be developed from multiscale concepts. As a result, conservative stabilised finite element procedures are presented for the advection-diffusion and incompressible Navier-Stokes equations.