Devising HDGHDG methods for Stokes flow: An overview

• We review the recent development of the HDGHDG methods for Stokes flow.
• We show how to devise the methods and discuss their different formulations.
• We describe how to systematically construct superconvergent methods.
• We describe how to obtain globally, divergence-free approximate velocities.
• We discuss extensions and several open problems.

We provide a short overview of our recent work on the devising of hybridizable discontinuous Galerkin (HDGHDG) 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 HDGHDG methods by using, as building blocks, the local spaces of superconvergent HDGHDG 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 HDGHDG methods with HH(div)-conforming velocity spaces, and how to extend the methods to other related systems. Several open problems are described.