A Brinkman-Penalized Finite Element Method for the Boussinesq Equations with Immersed Rigid Obstacles Under Uncertain Obstacle Motion

The Boussinesq equations are widely used to model incompressible thermally driven flows, but their numerical simulation becomes more challenging in domains containing fixed or moving rigid obstacles, particularly when the prescribed obstacle motion is uncertain. In this work, we develop a Brinkman-type fictitious-domain formulation for the incompressible Boussinesq system on a fixed computational domain. Brinkman penalization is used to impose the prescribed solid velocity, while a thermal penalty term enforces the temperature inside the immersed bodies. The resulting problem is discretized by a finite element method using skew-symmetric convective forms and grad-div stabilization. The uncertainty in the obstacle motion is introduced through a random oscillation amplitude and treated by a non-intrusive stochastic collocation method based on Gauss–Legendre quadrature. Stability and convergence of the fully discrete scheme are established. Numerical experiments for moving heated and hot–cold obstacles demonstrate the influence of the uncertain oscillation amplitude on the Nusselt number, kinetic energy, mean vorticity, and the spatial distributions of the mean and standard deviation of the solution fields. The results indicate that the proposed method provides a stable numerical framework for simulating incompressible thermally coupled flows with moving immersed obstacles under uncertainty in the prescribed obstacle motion.