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

Pietro Benedusi,Paola Ferrari,Rolf Krause,Carlo Garoni,Stefano Serra-Capizzano

doi:10.1007/s10915-021-01567-z

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.

Highlights

In recent years, with ever increasing computational capacities, space-time methods have received fast growing attention from the scientific community
We focus on the diffusion equation
Each preconditioned GMRES (PGMRES) iteration requires solving a linear system with coefficient matrix given by the preconditioner PN[q,np,k](K ), and this is not required in a GMRES iteration

Summary

Introduction

With ever increasing computational capacities, space-time methods have received fast growing attention from the scientific community. As it is known, each PGMRES iteration requires solving a linear system with coefficient matrix given by the preconditioner PN[q,,np,k](K ), and this is not required in a GMRES iteration. When solving the N (q + 1) linear systems with matrix Kn,[ p](K ) occurring at each PGMRES iteration, it is enough to approximate their solutions by performing only a few standard multigrid iterations in order to achieve an excellent PGMRES run-time; and, only one standard multigrid iteration is sufficient. We illustrate through numerical experiments the performance of the proposed solver and we compare it to the performance of other benchmark parallel solvers, such as the PGMRES with block-wise ILU(0) preconditioner

Implementation Details

Experimental Setting

Iteration Count and Timing

Scaling

Conclusions