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.
翻译:对在[B\'urger, M\'endez, Ruiz-Baier, SINUM (2019年), 57:1318-1343 中采用的双扩散模型和$\mathbf{H}(\mathrm{div}div})美元匹配方法的分析扩大到有时间依赖的情况,此外,对基于残余的后继差差错测算器的效率和可靠性的分析也扩大到稳定、半分解和完全离散的问题。因此采用的方法模拟盐水驱动流中小粒子的沉积。该方法包括速度和相容断裂式压力的Brezzi-Douglas-Marini近似值,而拉格朗贾元素用于浓度和盐度分布。数字测试证实了拟议方案组合和适应战略的特性,该组合和适应战略的受平流后差差误指标指导。