Adaptive domain decomposition method for time-dependent problems with applications in fluid dynamics

Abstract

We deal with the numerical solution of the time-dependent partial differential equations using the adaptive space-time discontinuous Galerkin (DG) method. The discretization leads to a nonlinear algebraic system at each time level, the size of the system is varying due to mesh adaptation. A Newton-like iterative solver leads to a sequence of linear algebraic systems which are solved by GMRES solver with a domain decomposition preconditioner. Particularly, we consider additive and hybrid two-level Schwarz preconditioners which are efficient and easy to implement for DG discretization. We study the convergence of the linear solver in dependence on the number of subdomains and the number of element of the coarse grid. We propose a simplified cost model measuring the computational costs in terms of floating-point operations, the speed of computation, and the wall-clock time for communications among computer cores. Moreover, the cost model serves as a base of the presented adaptive domain decomposition method which chooses the number of subdomains and the number of element of the coarse grid in order to minimize the computational costs. The efficiency of the proposed technique is demonstrated by two benchmark problems of compressible flow simulations.