Error estimates of a linear decoupled Euler–FEM scheme for a mass diffusion model

Abstract

We present error estimates of a linear fully discrete scheme for a three-dimensional mass diffusion model for incompressible fluids (also called Kazhikhov–Smagulov model). All unknowns of the model (velocity, pressure and density) are approximated in space by C 0-finite elements and in time an Euler type scheme is used decoupling the density from the velocity–pressure pair. If we assume that the velocity and pressure finite-element spaces satisfy the inf–sup condition and the density finite-element space contains the products of any two discrete velocities, we first obtain point-wise stability estimates for the density, under the constraint lim(h,k)→0 h/k = 0 (h and k being the space and time discrete parameters, respectively), and error estimates for the velocity and density in energy type norms, at the same time. Afterwards, error estimates for the density in stronger norms are deduced. All these error estimates will be optimal (of order $${\mathcal{O}(h+k)}$$ ) for regular enough solutions without imposing nonlocal compatibility conditions at the initial time. Finally, we also study two convergent iterative methods for the two problems to solve at each time step, which hold constant matrices (independent of iterations).