Efficient Algebraic Two-Level Schwarz Preconditioner for Sparse Matrices

Domain decomposition methods are among the most efficient for solving sparse linear systems of equations. Their effectiveness relies on a judiciously chosen coarse space. Originally introduced and theoretically proved to be efficient for self-adjoint operators, spectral coarse spaces have been proposed in the past few years for indefinite and non-self-adjoint operators. This paper presents a new spectral coarse space that can be constructed in a fully-algebraic way unlike most existing spectral coarse spaces. We present theoretical convergence result for Hermitian positive definite diagonally dominant matrices. Numerical experiments and comparisons against state-of-the-art preconditioners in the multigrid community show that the resulting two-level Schwarz preconditioner is efficient especially for non-self-adjoint operators. Furthermore, in this case, our proposed preconditioner outperforms state-of-the-art preconditioners

Space-Time Block Preconditioning for Incompressible Resistive Magnetohydrodynamics

This work develops a novel all-at-once space-time preconditioning approach for resistive magnetohydrodynamics (MHD), with a focus on model problems targeting fusion reactor design. We consider parallel-in-time due to the long time domains required to capture the physics of interest, as well as the complexity of the underlying system and thereby computational cost of long-time integration. To ameliorate this cost by using many processors, we thus develop a novel approach to solving the whole space-time system that is parallelizable in both space and time. We develop a space-time block preconditioning for resistive MHD, following the space-time block preconditioning concept first introduced by Danieli et al. in 2022 for incompressible flow, where an effective preconditioner for classic sequential time-stepping is extended to the space-time setting. The starting point for our derivation is the continuous Schur complement preconditioner by Cyr et al. in 2021, which we proceed to generalise in order to produce, to our knowledge, the first space-time block preconditioning approach for the challenging equations governing incompressible resistive MHD. The numerical results are promising for the model problems of island coalescence and tearing mode, with the overhead computational cost associated with space-time preconditioning versus sequential time-stepping being modest and primarily in the range of 2x-5x, which is low for parallel-in-time schemes in general. Additionally, the scaling results for inner (linear) and outer (nonlinear) iterations are flat in the case of fixed time-step size and only grow very slowly in the case of time-step refinement.Comment: 25 pages, 4 figures, 3 table