Monolithic Multigrid for a Reduced-Quadrature Discretization of Poroelasticity
Advanced finite-element discretizations and preconditioners for models of poroelasticity have attracted significant attention in recent years. The equations of poroelasticity offer significant challenges in both areas, due to the potentially strong coupling between unknowns in the system, saddle-point structure, and the need to account for wide ranges of parameter values, including limiting behavior such as incompressible elasticity. This paper was motivated by an attempt to develop monolithic multigrid preconditioners for the discretization developed in [C. Rodrigo et al., Comput. Methods App. Mech. Engrg, 341 (2018), pp. 467--484]; we show here why this is a difficult task and, as a result, we modify the discretization in [Rodrigo et al.] through the use of a reduced-quadrature approximation, yielding a more “solver-friendly” discretization. Local Fourier analysis is used to optimize parameters in the resulting monolithic multigrid method, allowing a fair comparison between the performance and costs of methods based on Vanka and Braess--Sarazin relaxation. Further, numerical results are presented to validate the local Fourier analysis predictions and demonstrate efficiency of the algorithms. Finally, a comparison to existing block-factorization preconditioners is also given.