Space-time domain decomposition methods and a posteriori error estimates for the heat equation

Sarah Ali Hassan (1, 2) , Caroline Japhet (3) , Michel Kern (1, 2) , Martin Vohralík (1, 2)

1 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
3 LAGA - Laboratoire Analyse, Géométrie et Applications

This paper develops a posteriori estimates for global-in-time, nonoverlapping domain decomposition (DD) methods for heterogeneous diffusion problems. The method uses optimized Schwarz waveform relaxation (OSWR) with Robin transmission conditions on the space-time interface between subdomains, and a lowest-order Raviart–Thomas–Nédélec discretization in the subdomains. Our estimates yield a guaranteed and fully computable upper bound on the error measured in the space-time energy norm of [19, 20], at each iteration of the space-time DD algorithm, where the spatial discretization, the time discretization, and the domain decomposition error components are estimated separately. Thus, an adaptive space-time DD algorithm is proposed, wherein the iterations are stopped when the domain decomposition error does not affect significantly the global error, allowing important savings in terms of the number of domain decomposition iterations while guaranteeing a user-given precision. Numerical results for a two-dimensional heat equation are presented to illustrate the efficiency of our a posteriori estimates and the performance of the adaptive stopping criteria for the space-time DD algorithm.

Mots clés: mixed finite element method; space-time domain decomposition; Heterogeneous diffusion; discontinuous Galerkin in time; optimized Robin transmission conditions; a posteriori error estimate; stopping criteria

Domaines: Analyse numérique [math.NA]

Sarah Ali Hassan, Caroline Japhet, Michel Kern, Martin Vohralík. Space-time domain decomposition methods and a posteriori error estimates for the heat equation. EDP-NORMANDIE 2017 - VIe Colloque Edp-normandie, Oct 2017, Caen, France. pp.1-18. ⟨hal-01702428⟩