The discontinuous Petrov-Galerkin (DPG) finite element methodology first proposed in 2009 by Demkowicz and Gopalakrishnan—and subsequently developed by many others—offers a fundamental framework for developing robust residual-minimizing finite element methods, even for equations that usually cause problems for standard methods, such as convection-dominated diffusion and the Stokes equations. For a very broad class of well-posed problems, DPG offers provably optimal convergence rates with a modest convergence constant. In some of our experiments, DPG not only achieves the optimal rates, but gets extremely close to the best approximation available in the discrete space. Moreover, DPG provides a way to measure the error in the approximate solution, which can then robustly drive adaptivity.

Camellia is a robust, flexible software framework for DPG research and experimentation, built atop Trilinos. At present, Camellia supports 2D meshes of triangles and quads of variable polynomial order, provides mechanisms for easy specification of DPG variational forms, supports h- and p-refinements, and supports distributed computation of the stiffness matrix, among other features. By defining classes that capture the core conceptual kernels of DPG (and those of finite element methods generally), Camellia allows extremely rapid development of DPG solvers. For example, the entire specification of a Stokes bilinear form can be accomplished in just 45 lines of easily understood code.

In this presentation, I will introduce the salient features of DPG and Camellia, and their application to incompressible flow problems modeled by the Stokes and Navier-Stokes equations.