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We construct a fully discrete numerical scheme for three-dimensional incompressible fluids with mass diffusion (in density-velocity-pressure formulation), also called the Kazhikhov-Smagulov model. We will prove conditional stability and convergence, by using at most $C^0$-finite elements, although the density of the limit problem will have $H^2$-regularity. The key idea of our argument is first to obtain pointwise estimates for the discrete density by imposing the constraint $\lim_{(h,k)\to 0}h/k=0$ on the time and space parameters $(k,h)$. Afterwards, under the same constraint on the parameters, strong estimates for the discrete density in $l^\infty(H^1)$ and for the discrete Laplacian of the density in $l^2(L^2)$ are obtained. From here, the compactness and convergence of the scheme can be concluded with similar arguments as we used in [Math. Comp., to appear], where a different scheme is studied for two-dimensional domains which is unconditionally stable and convergent. Moreover, we study the asymptotic behavior of the numerical scheme as the diffusion parameter $\lambda$ goes to zero, obtaining convergence as $(k,h,\lambda)\to 0$ towards a weak solution of the density-dependent Navier-Stokes system provided that the constraint $\lim_{(\lambda,h,k)\to 0}h/(\lambda^2 k)=0$ on $(h,k,\lambda)$ is satisfied.

Conditional Stability and Convergence of a Fully Discrete Scheme for Three-Dimensional Navier–Stokes Equations with Mass Diffusion