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

Elliptic partial differential equations must be solved numerically for many problems in numerical relativity, such as initial data for every simulation of merging black holes and neutron stars. Existing elliptic solvers can take multiple days to solve these problems at high resolution and when matter is involved, because they are either hard to parallelize or require a large amount of computational resources. Here we present a new solver for linear and non-linear elliptic problems that is designed to scale with resolution and to parallelize on computing clusters. To achieve this we employ a discontinuous Galerkin discretization, an iterative multigrid-Schwarz preconditioned Newton-Krylov algorithm, and a task-based parallelism paradigm. To accelerate convergence of the elliptic solver we have developed novel subdomain-preconditioning techniques. We find that our multigrid-Schwarz preconditioned elliptic solves achieve iteration counts that are independent of resolution, and our task-based parallel programs scale over 200 million degrees of freedom to at least a few thousand cores. Our new code solves a classic black-hole binary initial-data problem faster than the spectral code SpEC when distributed to only eight cores, and in a fraction of the time on more cores. It is publicly accessible in the next-generation SpECTRE numerical relativity code. Our results pave the way for highly-parallel elliptic solves in numerical relativity and beyond.Comment: 24 pages, 18 figures. Results are reproducible with the ancillary input file

Preconditioned implicit time integration schemes for Maxwell’s equations on locally refined grids

In this paper, we consider an efficient implementation of higher-order implicit time integration schemes for spatially discretized linear Maxwell’s equations on locally refined meshes. In particular, our interest is in problems where only a few of the mesh elements are small while the majority of the elements is much larger. We suggest to approximate the solution of the linear systems arising in each time step by a preconditioned Krylov subspace method, e.g., the quasi-minimal residual method by Freund and Nachtigal [13]. Motivated by the analysis of locally implicit methods by Hochbruck and Sturm [25], we show how to construct a preconditioner in such a way that the number of iterations required by the Krylov subspace method to achieve a certain accuracy is bounded independently of the diameter of the small mesh elements. We prove this behavior by using Faber polynomials and complex approximation theory. The cost to apply the preconditioner consists of the solution of a small linear system, whose dimension corresponds to the degrees of freedom within the fine part of the mesh (and its next coarse neighbors). If this dimension is small compared to the size of the full mesh, the preconditioner is very efficient. We conclude by verifying our theoretical results with numerical experiments for the fourth-order Gauß-Legendre Runge–Kutta method

Can coercive formulations lead to fast and accurate solution of the Helmholtz equation?

A new, coercive formulation of the Helmholtz equation was introduced in [Moiola, Spence, SIAM Rev. 2014]. In this paper we investigate $h$-version Galerkin discretisations of this formulation, and the iterative solution of the resulting linear systems. We find that the coercive formulation behaves similarly to the standard formulation in terms of the pollution effect (i.e. to maintain accuracy as $k\to\infty$, $h$ must decrease with $k$ at the same rate as for the standard formulation). We prove $k$-explicit bounds on the number of GMRES iterations required to solve the linear system of the new formulation when it is preconditioned with a prescribed symmetric positive-definite matrix. Even though the number of iterations grows with $k$, these are the first such rigorous bounds on the number of GMRES iterations for a preconditioned formulation of the Helmholtz equation, where the preconditioner is a symmetric positive-definite matrix.Comment: 27 pages, 7 figure

Schnelle Löser für Partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

A geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing

We present a parallel data structure which is directly linked to geometric quantities of an underlying mesh and which is well adapted to the requirements of a general finite element realization. In addition, we define an abstract linear algebra model which supports multigrid methods (extending our previous work in Comp. Vis. Sci. 1 (1997) 27-40). Finally, we apply the parallel multigrid preconditioner to several configurations in linear elasticity and we compute the condition number numerically for different smoothers, resulting in a quantitative evaluation of parallel multigrid performance