The discontinuous Petrov-Galerkin finite element methodology of Demkowicz and Gopalakrishnan (DPG) [1, 2] offers a host of appealing features, including automatic stability and minimization of the residual in a user-controllable energy norm. DPG is, moreover, well-suited for high-performance computing, in that the extra work required by the method is embarrassingly parallel; the use of a discontinuous test space allows the computation of optimal test functions to be done element-wise. Additionally, the approach gives almost total freedom in the choice of basis functions, so that high-order discretizations can be employed to increase computational intensity (the number of floating point operations per unit of communication). Finally, since the method is stable even on a coarse mesh and comes with a built-in error measurement, it enables robust adaptivity which in turn means less human involvement in the solution process, a desirable feature when running large-scale computations.

Camellia [3] is a software framework for DPG with the aim of enabling rapid development of DPG solvers both for running on a laptop and at scale. Camellia supports spatial meshes in 1D through 3D; initial support for space-time elements is also available. Camellia supports h- and p-adaptivity, and offers distributed computation of essentially all the algorithmic components of a DPG solve. (One exception, which we plan to address, is the generation and storage of the mesh geometry; at present, this happens redundantly on each MPI rank.) Camellia supports static condensation for reduction of the global problem, and has a robust, flexible interface for using third-party direct and iterative solvers for the global solve.

Until recently, we have almost always solved the global DPG system matrix using parallel direct solvers such as SuperLU Dist. This is not a scalable strategy, particularly for 3D and space-time meshes. Both memory and time costs therefore motivate the present work, an exploration of iterative solvers in the context of Poisson and Stokes problems. Since Camellia's adaptive mesh hierarchy provides us with rich geometric information, we focus on hp-geometric multigrid preconditioners with additive Schwarz smoothers of minimal or small overlap. Preconditioning a conjugate gradient solve using such preconditioners, we are able to solve much larger problems within the same memory footprint.

References
[1] L. Demkowicz and J. Gopalakrishnan. A class of discontinuous Petrov-Galerkin methods. Part I : The transport equation. Comput. Methods Appl. Mech. Engrg., 199:1558-1572, 2010. See also ICES Report 2009-12.
[2] L. Demkowicz and J. Gopalakrishnan. A class of discontinuous Petrov-Galerkin methods. Part II: Optimal test functions. Numer. Meth. Part. D. E., 27(1):70-105, January 2011.
[3] N. V. Roberts. Camellia: A software framework for discontinuous Petrov-Galerkin methods. Computers & Mathematics with Applications, 68(11):1581-1604, December 2014.