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

TL;DR

This paper extends divergence-conforming methods for transient doubly-diffusive flows, providing error analysis and demonstrating their effectiveness in simulating sedimentation in salinity-driven flows.

Contribution

It develops and analyzes a divergence-conforming finite element method for time-dependent doubly-diffusive flows, including a posteriori error estimators and adaptive strategies.

Findings

Numerical tests confirm the accuracy and efficiency of the proposed methods.

The adaptive strategy effectively guides mesh refinement based on error indicators.

The method accurately simulates sedimentation in salinity-driven flows.

Abstract

The analysis of the double-diffusion model and $\mathbf{H}(\mathrm{div})$-conforming method introduced in [B\"urger, M\'endez, Ruiz-Baier, SINUM (2019), 57:1318--1343] is extended to the time-dependent case. In addition, the efficiency and reliability analysis of residual-based {\it a posteriori} error estimators for the steady, semi-discrete, and fully discrete problems is 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 {\it a posteriori} error indicators.