We consider a coupled nonlinear system of equations that describe unsaturated flow in heterogeneous poroelastic media. For the numerical solution, we use a finite-element approximation in space and present an efficient multiscale two-grid solver for solving the coupled system of equations. The proposed two-grid solver contains two main parts: (i) an accurate coarse-grid approximation based on local spectral spaces, and (ii) coupled smoothing iterations based on an overlapping multiscale Vanka method. A Vanka smoother and local spectral coarse grids come with significant computational cost in the setup phase. To avoid constructing a new solver for each time step and/or nonlinear iteration, we utilize an implicit–explicit integration scheme in time, where we partition the nonlinear operator as a sum of linear and nonlinear parts. In particular, we construct an implicit linear approximation of the stiff components that remains fixed across all time, while treating the remaining nonlinear residual explicitly. This allows us to construct a robust two-grid solver offline and utilize it for fast and efficient online time integration. A linear stability analysis of the proposed novel coupled scheme is presented based on the representation of the system as a two-step scheme. We show that the careful decomposition of linear and nonlinear parts guarantees a linearly stable scheme. A numerical study is presented for a nonlinear coupled test problem of unsaturated flow in heterogeneous poroelastic media. We demonstrate the robustness of the two-grid solver, particularly the efficacy of block smoothing compared with simple pointwise smoothing, and illustrate the accuracy and stability of implicit–explicit time integration.
Vasilyeva et al. (Thu,) studied this question.