Hybrid Discontinuous Galerkin methods

The concept

hdg — In HDG methods additional facet unknowns are introduced to reduce element neighbor couplings.

The key idea of Hybrid Discontinuous Galerkin (HDG) methods is to effectively obtain the flexibility benefits of Discontinuous Galerkin (DG) methods with a reduced computational overhead. Discontinuous Galerkin (DG) methods are often criticized for the additional degrees of freedoms stemming from splitting up continuous functions. More importantly - when solving linear systems - element unknowns couple to neighboring element unknowns which introduces a lot of entries in the system matrix. In HDG methods, although even more unknowns are introduced on the skeleton of the mesh, the coupling between element unknowns is reduced to couplings between element unknowns and facet unknowns. The coupling between neighboring elements still exists, but is now indirect through the facet unknowns. This is especially benefitial for higher order discretizations as facet unknowns are of lower dimensional (\(\mathcal{O}(p^{d-1})\) compared to \(\mathcal{O}(p^{d})\) element unknowns).

Tweaks

In the case of diffusion dominated elliptic operators an additional trick can significantly improve the performance of HDG methods by reducing the polynomial degree of the facet unknowns. This trick is also known as "projected jumps" and is similar to the "post-processing" step in hybrid mixed methods. We introduced this trick in a remark in [Leh10] and discussed efficient and flexible implementations in [LS16].

Applications

We use HDG methods whenever we want to make use of the benefits of DG methods. The following two are the predominant cases where we consider HDG methods:

Convection dominated problems as (H)DG discretizations allow for Upwinding schemes and
structure-preserving discretizations with (only) \(H(\operatorname{div})\)-conforming finite elements for incompressible flows (with solutions in \(H^1\)).

Application areas where we applied HDG methods so far are:

incompressible Stokes and Navier-Stokes, cf. [Leh10, LS16, LLS18, LLS19, FKL+19, LLS20],
convection-diffusion equations, cf. [Leh10],
Darcy-Stokes coupling, cf. [FL],
linear elasticity, cf. [FLLS20],
helioseismology, cf.

Linear systems

A natural approach for preconditioning HDG discretized schemes is BDDC preconditiong. BDDC for HDG discretizations is introduced (and analyzed) in this talk (slides) by Christoph Lehrenfeld from 2012, cf. [SL13].

Getting started

If you are new to HDG methods, we recommend to read the first chapters of [Leh10] and the corresponding section in the NGSolve i-tutorials.

References: [ Leh10, SL13, LS16, FL, LLS18, LLS19, FKL+19, LLS20, FLLS20 ]