Divergence-conforming methods for transient double-diffusive flows: a priori and a posteriori error analysis

The analysis of an $\textbf {H}(\textrm {div})$-conforming method for a model of double-diffusive flow in porous media introduced in Burger, Mendez & Ruiz-Baier (2019, On H(div)-conforming methods for double-diffusion equations in porous media. SIAM J. Numer. Anal., 57,1318-1343) is extended to the time-dependent case. In addition, the efficiency and reliability of residual-based a posteriori error estimators for the steady, semidiscrete and fully discrete problems are established. The resulting methods are applied to simulate the sedimentation of small particles in salinity-driven flows. The method consists of Brezzi-Douglas-Marini approximations for velocity and compatible piecewise discontinuous pressures, whereas Lagrangian elements are used for concentration and salinity distribution. Numerical tests confirm the properties of the proposed family of schemes and of the adaptive strategy guided by the a posteriori error indicators.