Typical solutions of incompressible flow problems involve both fine- and large-scale phenomena, so that a uniform finite element mesh of sufficient granularity is at best wasteful of computational resources, and at worst infeasible because of resource limitations. Thus, adaptive mesh refinements are desirable. In industry, the adaptivity schemes used are ad hoc, requiring a domain expert to predict features of the solution. A badly chosen mesh may cause the code to take considerably longer to converge, or fail to converge altogether. Typically, the Navier-Stokes solve is just one component in an optimization loop, which means that failures requiring human intervention are costly.

Our aim, therefore, is to develop a solver for the incompressible Navier-Stokes equations that provides robust adaptivity starting from a coarse mesh. By robust, we mean both that the solver always converges to a solution in predictable time, and that the adaptive scheme is independent of the problem---no special expertise is required for adaptivity.

The cornerstone of our approach is the discontinuous Petrov-Galerkin (DPG) finite element methodology of Demkowicz and Gopalakrishnan [1,2]. Whereas Bubnov-Galerkin methods use the same function space for both test and trial functions, Petrov-Galerkin methods allow the spaces for test and trial functions to differ. In DPG, the test functions are computed on the fly so that the solution residual is minimized in the energy norm. For a very broad class of well-posed problems, DPG offers provably optimal convergence rates with a modest stability constant - the "inf-sup" constants governing the convergence are mesh-independent, and of the same order as those governing the continuous problem [5]. In some of our experiments, DPG not only achieves the optimal rates, but gets very close to the best solution available in the discrete space. DPG also provides an accurate mechanism for measuring the error, and this can be used to drive adaptive mesh refinements.

Central to our study of these problems has been the use and further development of Camellia [4], a toolbox we developed for solving DPG problems which uses Sandia's Trilinos library of packages [3]. At present, Camellia supports 2D meshes of triangles and quads of variable polynomial order, provides mechanisms for rapid specification of DPG variational forms, supports h- and p- refinements, and supports distributed computation of the stiffness matrix, among other features. We are adding support for meshes of arbitrary spatial dimension, space-time elements, and distributed mesh and solution representation.

In this presentation, I will introduce salient features of DPG and Camellia, and their application to several 2D incompressible flow problems modeled by the Navier-Stokes equations, including the lid-driven cavity flow problem and the flow past a cylinder problem.

[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] M.A. Heroux, et al. An overview of the Trilinos project. ACM Trans. Math. Softw., 31(3):397-423, 2005.
[4] Nathan V. Roberts. Camellia: A software framework for discontinuous Petrov-Galerkin methods. Computers & Mathematics with Applications, 2014 (submitted).
[5] Nathan V. Roberts, Tan Bui-Thanh, and Leszek F. Demkowicz. The DPG method for the Stokes problem. Computers & Mathematics with Applications, 2014.