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We prove the convergence of an incremental projection numerical scheme forthe time-dependent incompressible Navier--Stokes equations, without anyregularity assumption on the weak solution. The velocity and the pressure arediscretised in conforming spaces, whose the compatibility is ensured by theexistence of an interpolator for regular functions which preserves approximatedivergence free properties. Owing to a priori estimates, we get the existenceand uniqueness of the discrete approximation. Compactness properties are thenproved, relying on a Lions-like lemma for time translate estimates. It is thenpossible to show the convergence of the approximate solution to a weak solutionof the problem. The construction of the interpolator is detailed in the case ofthe lowest degree Taylor-Hood finite element.

Convergence of the incremental projection method using conforming approximations