Monolithic multigrid for higher-order discretizations of poroelasticity

Mathematical models of poroelasticity, the study of the behaviour of uid-saturated porous media, present complex challenges in numerical simulation due to their inherent coupling between uid and solid phases. In this study, we propose higher-order discretization techniques for poroelasticity problems, that we couple with monolithic multigrid methods to enable efficient high-fidelity simulations. These discretizations are based on higher-order finite elements in space (including reduced quadrature techniques to effectively model nearly incompressible solid phases) and implicit Runge- Kutta methods in time, to ensure robustness and stability in the time-stepping procedure. The monolithic multigrid approach leverages recent work extending Vanka-style relaxation to incompressible ow models, that we adapt to the equations of poroelasticity. Through numerical experiments and comparisons, we demonstrate the effectiveness of our proposed approach in accurately capturing the behaviour of poroelastic models while maintaining computational efficiency.