Multigrid Reduced in Time for Isogeometric Analysis

[EN] Isogeometric Analysis (IgA) can be seen as the natural extension of the Finite Element Method (FEM) to high-order B-spline basis functions. Combined with a time integration scheme within the method of lines, IgA has become a viable alternative to FEM for time-dependent problems. However, as processors’ clock speeds are no longer increasing but the number of cores are going up, traditional (i.e., sequential) time integration schemes become more and more the bottleneck within these large-scale computations. The Multigrid Reduced in Time (MGRIT) method is a parallel-in-time integration method that enables exploitation of parallelism not only in space but also in the temporal direction. In this paper, we apply MGRIT to discretizations arising from IgA for the first time in the literature. In particular, we investigate the (parallel) performance of MGRIT in this context for a variety of geometries, MGRIT hierarchies and time integration schemes. Numerical results show that the MGRIT method converges independent of the mesh width, spline degree of the B-spline basis functions and time step size ∆t and is highly parallelizable when applied in the context of IgA.Tielen, R.; Möller, M.; Vuik, K. (2022). Multigrid Reduced in Time for Isogeometric Analysis. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 47-56. https://doi.org/10.4995/YIC2021.2021.12219OCS475

h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

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Multilevel convergence analysis of multigrid-reduction-in-time

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A multigrid perspective on the parallel full approximation scheme in space and time

For the numerical solution of time-dependent partial differential equations, time-parallel methods have recently shown to provide a promising way to extend prevailing strong-scaling limits of numerical codes. One of the most complex methods in this field is the "Parallel Full Approximation Scheme in Space and Time" (PFASST). PFASST already shows promising results for many use cases and many more is work in progress. However, a solid and reliable mathematical foundation is still missing. We show that under certain assumptions the PFASST algorithm can be conveniently and rigorously described as a multigrid-in-time method. Following this equivalence, first steps towards a comprehensive analysis of PFASST using block-wise local Fourier analysis are taken. The theoretical results are applied to examples of diffusive and advective type