- Francisco Gaspar - Multigrid waveform relaxation. Application to the time-fractional heat equation.
The numerical solution of large linear systems arising from the discretization of partial differential equations with evolutionary behavior is of interest in many applications. When dealing with such time-dependent problems, the classic approach (time-stepping) is based on solving sequentially the problem for one time step after the other. However, this strategy does not allow the parallelization of the temporal variable. In many practical problems very large number of time-steps are required, what increases the computational cost of the simulations. In order to overcome this, one can increase the concurrency by using time-parallel and full space-time methods. Multigrid methods are known to have optimal complexity for solving many numerical problems. The use of multigrid for adding parallelism to time integration allows for faster time-to-solution in comparison with classical time-stepping approaches. The aim of this work is the analysis of the performance of the multigrid waveform relaxation for the time-fractional heat equation.
- Peiyao Luo - Monolithic Multigrid Method for the Coupled Stokes Flow and Deformable Porous Medium System
The interaction between fluid flow and a deformable porous medium is a complicated multi-physics problem which can be described by a coupled model based on the Stokes and poroelastic equations. A monolithic multigrid method together with a coupled Vanka smoother and a decoupled Uzawa smoother is employed as an efficient numerical technique for the linear discrete system obtained by finite volumes on staggered grids. A novelty in our modeling approach is that at the interface of the fluid and poroelastic medium, two unknowns from the different subsystems are defined at the same grid point. We propose a special discretization at or near the points in the interface, which combines the approximation of the governing equations and the considered interface conditions. In the decoupled Uzawa smoother, Local Fourier analysis (LFA) helps us to select optimal values of the relaxation parameter appearing. To implement the monolithic multigrid method, grid partitioning is used to deal with the interface updates when communications are required between two subdomains. Numerical experiments show that the proposed numerical method provides an excellent convergence. The efficiency and robustness of the method are confirmed in numerical experiments with typically small values of the physical coefficients.