We provide a short overview of our recent work on the devising of hybridizable discontinuous Galerkin ( HDG) methods for the Stokes equations of incompressible flow. First, we motivate and display the general form of the methods and show that they provide a well defined approximate solution for arbitrary polyhedral elements. We then discuss three different but equivalent formulations of the methods. Next, we describe a systematic way of constructing superconvergent HDG methods by using, as building blocks, the local spaces of superconvergent HDG methods for the Laplacian operator. This can be done, so far, for simplexes, parallelepipeds and prisms. Finally, we show how, by means of an elementwise computation, we can obtain divergence-free velocity approximations converging faster than the original velocity approximation when working with simplicial elements. We end by briefly discussing other versions of the methods, how to obtain HDG methods with H(div)-conforming velocity spaces, and how to extend the methods to other related systems. Several open problems are described.
Bibliographical noteFunding Information:
The first author would like to acknowledge the partial support of the National Science Foundation (Grant No. DMS-1115331 ) and of the University of Minnesota Supercomputing Institute.
- Divergence-free approximations
- Hybridizable discontinuous galerkin methods
- Stokes equations
- Unstructured meshes