In this work, a multigrid waveform relaxation method is proposed for solving a collocated finite difference discretization of the linear Biot’s model. This gives rise to the first space–time multigrid solver for poroelasticity equations in the literature. The waveform relaxation iteration is based on a point-wise Vanka smoother that couples the pressure variable at a grid-point with the displacements around it. A semi-algebraic mode analysis is proposed to theoretically analyze the convergence of the multigrid waveform relaxation algorithm. This analysis is novel since it combines the semi-algebraic analysis, suitable for parabolic problems, with the non-standard analysis for overlapping smoothers. The practical utility of the method is illustrated through several numerical experiments in one and two dimensions.

Additional Metadata
Keywords Multigrid waveform relaxation, Poroelasticity, Semi-algebraic mode analysis, Space–time grids, Vanka smoother
Persistent URL dx.doi.org/10.1007/s40314-018-0603-9
Journal Computational and Applied Mathematics
Citation
Franco, S.R, Rodrigo, C, Gaspar, F.J, & Pinto, M.A.V. (2018). A multigrid waveform relaxation method for solving the poroelasticity equations. Computational and Applied Mathematics, 37(4), 4805–4820. doi:10.1007/s40314-018-0603-9