TAYLOR–HOOD FINITE ELEMENTS FOR STOKES FLOW WITH VARIABLE DENSITY: BAND-CONDITIONED WELL-POSEDNESS AND OPTIMAL CONVERGENCE

We analyze the Taylor–Hood finite element discretization of a Stokes-type model for a compressible flow with variable density, in which mass conservation is replaced by div(ρu) = 0 with ρ a smooth positive density. The continuous problem is well-posed under an explicit band condition, which relates the height of a band containing the domain to the ratio between the lower bound of the density and the sup-norm of its gradient. The same condition controls both the coercivity of the velocity form and the inf-sup condition of the mixed form. We then study the Taylor–Hood P2/P1 discretization: we prove discrete stability, and we derive an a priori error estimate of order O(h3) in L2 for the velocity, O(h2) in H1 for the velocity, and O(h2) in L2 for the pressure. These are the optimal rates, and the variable density leaves them intact whenever the band condition holds. Numerical experiments on a variable-density manufactured solution confirm the rates. We close with a residual-based a posteriori estimator, with proofs of reliability and efficiency up to data oscillation, and with a numerical validation of the estimator on an L-shaped domain. Key words. variable-density Stokes flow, Taylor–Hood elements, band condition, inf-sup, optimal convergence AMS subject classifications. 65N30, 65N12, 76D07, 76M10