No abstract

Please cite this article in press as: N.V. Roberts et al., A discontinuous Petrov-Galerkin methodology for adaptive solutions to the incompressible Navier-Stokes equations, J. Comput. Phys. (2015), http://dx. AbstractThe discontinuous Petrov-Galerkin methodology with optimal test functions (DPG) of Demkowicz and Gopalakrishnan [18,20] guarantees the optimality of the solution in an energy norm, and provides several features facilitating adaptive schemes. Whereas Bubnov-Galerkin methods use identical trial and test spaces, Petrov-Galerkin methods allow these function spaces to differ. In DPG, test functions are computed on the fly and are chosen to realize the supremum in the inf-sup condition; the method is equivalent to a minimum residual method. For well-posed problems with sufficiently regular solutions, DPG can be shown to converge at optimal rates-the inf-sup constants governing the convergence are mesh-independent, and of the same order as those governing the continuous problem [48]. DPG also provides an accurate mechanism for measuring the error, and this can be used to drive adaptive mesh refinements.We employ DPG to solve the steady incompressible Navier-Stokes equations in two dimensions, building on previous work on the Stokes equations, and focusing particularly on the usefulness of the approach for automatic adaptivity starting from a coarse mesh. We apply our approach to a manufactured solution due to Kovasznay as well as the lid-driven cavity flow, backward-facing step, and flow past a cylinder problems.

We explore a vexing benchmark problem for viscoelastic fluid flows with the discontinuous Petrov-Galerkin (DPG) finite element method of Demkowicz and Gopalakrishnan [1,2]. In our analysis, we develop an intrinsic a posteriori error indicator which we use for adaptive mesh generation. The DPG method is useful for the problem we consider because the method is inherently stable---requiring no stabilization of the linearized discretization in order to handle the advective terms in the model. Because stabilization is a pressing issue in these models, this happens to become a very useful property of the method which simplifies our analysis. This built-in stability at all length scales and the a posteriori error indicator additionally allows for the generation of parameter-specific meshes starting from a common coarse initial mesh. A DPG discretization always produces a symmetric positive definite stiffness matrix. This feature allows us to use the most efficient direct solvers for all of our computations. We use the Camellia finite element software package [3,4] for all of our analysis.Comment: 20 pages, 18 figures, 6 table

No abstract

The discontinuous Petrov-Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan [15,17] guarantees the optimality of the solution in an energy norm, and provides several features facilitating adaptive schemes. A key question that has not yet been answered in general-though there are some results for Poisson, e.g.-is how best to precondition the DPG system matrix, so that iterative solvers may be used to allow solution of large-scale problems.In this paper, we detail a strategy for preconditioning the DPG system matrix using geometric multigrid which we have implemented as part of Camellia [26], and demonstrate through numerical experiments its effectiveness in the context of several variational formulations. We observe that in some of our experiments, the behavior of the preconditioner is closely tied to the discrete test space enrichment.We include experiments involving adaptive meshes with hanging nodes for lid-driven cavity flow, demonstrating that the preconditioners can be applied in the context of challenging problems. We also include a scalability study demonstrating that the approach-and our implementation-scales well to many MPI ranks. Acknowledgments.convergence rates have either been proved a priori or observed in numerical experiments-in some cases, the solutions are even nearly optimal in terms of the absolute L 2 error (not merely the rate). The global system matrices that arise from DPG formulations are symmetric (Hermitian) positive definite, making them good candidates for solution using the conjugate gradient (CG) method. However, these matrices often have fairly large condition numbers which scale as 1 h 2 (see [20] for the scaling estimate, and Table 9.3 in [25] for measurements), so that a good preconditioner is required before CG can be used effectively.For us, a key motivation in the present work is the scalability of our solvers in Camellia [26]. Prior to developing the preconditioners presented here, direct solvers were almost exclusively employed. These solvers only scale to a certain limited system size, and can require substantially more memory than iterative solvers. This is of particular concern for high-performance computing systems, where the memory per core is increasingly limited-for example, Argonne's Mira supercomputer has a BlueGene/Q architecture that has just one gigabyte per core.The structure of the paper is as follows. In Section 2, we review some previous work in preconditioning DPG and similar systems. In Section 3, we briefly introduce the DPG methodology and state the variational formulations we use for our numerical experiments. In Section 4, we detail our geometric multigrid preconditioners. In Section 5, we present a wide variety of numerical experiments demonstrating the effectiveness of the approach. We examine the scalability of our implementation in Section 6. We conclude in Section 7. Some notes on our implementation in Camellia can be found in Appendix A; for reference, we provide numerical values for the smoother weights we employ in Appendix B. Literatu...