Solving the Mimetic Pressure Poisson Equation in a 2D Lock Exchange Flow Using DMDc–LQR–LGMRES

The numerical simulation of incompressible density driven flows governed by the Navier–Stokes equations remains a central problem in computational fluid dynamics, particularly when structure preserving discretizations are employed to maintain the geometric properties of differential operators. In this work, we build upon the two dimensional lock exchange benchmark implementation provided with the MOLE (Mimetic Operators Library Enhanced) framework, which simulates density-driven flow under the Boussinesq approximation. Within projection-based formulations, the incompressibility constraint leads to a pressure correction step in which the pressure field is obtained by solving a two-dimensional Poisson equation. In this context, we investigate a controlled Krylov subspace strategy based on the LGMRES iterative method, where the restart parameter is dynamically regulated through a data driven control framework combining Dynamic Mode Decomposition with control (DMDc) and Linear Quadratic Regulator (LQR) theory. This approach leads to the DMDc–LQR–LGMRES method, which is intended to mitigate stagnation effects commonly observed in restarted Krylov solvers. In addition, this strategy can be viewed as a prototyping step toward scalable High Performance Computing (HPC) implementations, where efficient and robust linear solvers play an important role in large-scale incompressible flow simulations.