hp-adaptive discontinuous Galerkin solver for elliptic equations in numerical relativity

Abstract

A considerable amount of attention has been given to discontinuous Galerkin methods for hyperbolic problems in numerical relativity, showing potential advantages of the methods in dealing with hydrodynamical shocks and other discontinuities. This paper investigates discontinuous Galerkin methods for the solution of elliptic problems in numerical relativity. We present a novel hp-adaptive numerical scheme for curvilinear and nonconforming meshes. It uses a multigrid preconditioner with a Chebyshev or Schwarz smoother to create a very scalable discontinuous Galerkin code on generic domains. The code employs compactification to move the outer boundary near spatial infinity. We explore the properties of the code on some test problems, including one mimicking Neutron stars with phase transitions. We also apply it to construct initial data for two or three black holes.

A scalable elliptic solver with task-based parallelism for the SpECTRE numerical relativity code

“…Nevertheless, the condition number can provide an indication for the expected rate of convergence. For the discontinuous Galerkin discretization we employ in this article, the condition number scales as κ ∝ p 2 /h, where p denotes a typical polynomial degree of the elements and h denotes a typical element size [39,40,49,50].…”

“…We employ a geometric V-cycle multigrid algorithm, as prototyped in Ref. [39]. 5 Our multigrid solver can be used stand-alone, or to precondition a Krylov-type linear solver as described in Section III B ("Krylov-accelerated multigrid").…”

“…This strategy follows Ref. [39] and ensures that coarse-grid field approximations always have an exact polynomial representation on finer grids.…”

“…In principle, the smoother can be any linear solver, including a Krylov-subspace solver as detailed in Section III B. However, to achieve good parallel performance we have developed a highly asynchronous additive Schwarz smoother [39,53,54], that we employ for preand post-smoothing on every level of the multigrid hierarchy. Note that we also apply it on the coarsest grid, where some authors apply a dedicated bottom smoother instead [55,56].…”

“…Our new elliptic solver in the SpECTRE code is based on the prototype presented in Ref. [39] and employs the discontinuous Galerkin discretization for generic elliptic equations developed in Ref. [40], which makes it applicable to a wide range of elliptic problems in numerical relativity and beyond.…”