Fast Parallel Solver for the Space-time IgA-DG Discretization of the Diffusion Equation

Abstract

We consider the space-time discretization of the diffusion equation, using an isogeometric analysis (IgA) approximation in space and a discontinuous Galerkin (DG) approximation in time. Drawing inspiration from a former spectral analysis, we propose for the resulting space-time linear system a multigrid preconditioned GMRES method, which combines a preconditioned GMRES with a standard multigrid acting only in space. The performance of the proposed solver is illustrated through numerical experiments, which show its competitiveness in terms of iteration count, run-time and parallel scaling.

Space-time multilevel Monte Carlo methods and their application to cardiac electrophysiology

“…Every linear problem in the form of ( 17) (arising at every Newton iteration) is solved by means of a space-time parallel GMRES with block Jacobi preconditioner. The spectral analysis of the space-time system in (17) motivates this choice, see [5] for all the details.…”

“…We apply state of the art methods in uncertainty quantification (UQ). This means that we combine space-time GMRES with a block Jacobi preconditioner [5] for solving the monodomain equation with multilevel quadrature methods for the UQ. In our practical implementation, we use the multilevel (quasi-) Monte Carlo method, compare [1,11,14,17,20].…”

An experimental comparison of a space-time multigrid method with PFASST for a reaction-diffusion problem

“…The first analysis on DG methods as time stepping techniques was provided by [12] and [17], followed by the work of [41,50,48]. More recently, specialized solution methods have been introduced, for example by [45,40,31,4]. A priori and posteriori error analysis have been also provided, e.g.…”

“…Specialized parallel solvers have been recently developed for large linear systems arising from space-time discretizations. We mention in particular the parallel STMG proposed by [24], the parallel preconditioners for space-time isogeometric analysis proposed by [28] and [4] as well as the block preconditioned GMRES by [37]. When dealing with a space-time discretization, where time is somehow considered as an additional spatial dimension, it is natural to extend the same paradigm for the solving process and consider space-time multigrid type algorithms.…”

“…For smoothing, we employ a GMRES solver, preconditioned with an incomplete LU factorization (PGMRES). More specialized preconditioners and corresponding tensor solvers could also be applied to system (3.4); see, e.g., [39,9,4]. If γ = 0, equation (3.4) is non-linear and the STMG algorithm is encapsulated in a Newton iteration.…”