Hybridizable Discontinuous Galerkin¶

Hybridizable Discontinuous Galerkin (HDG) introduces an independent trace unknown \(\hat{u}\) on the mesh skeleton. Element unknowns are solved locally in terms of \(\hat{u}\), and the global problem enforces flux continuity across faces.

For a first-order diffusion-type system,

\[ q + \nabla u = 0, \]

\[ \frac{\partial u}{\partial t} + \nabla \cdot F(u,q,w,v,x,t;\mu) = S(u,q,w,v,x,t;\mu), \]

HDG seeks element fields \((q_h,u_h)\) and a trace \(\hat{u}_h\) such that the element equations and face flux-continuity equations are satisfied.

Static Condensation¶

The core HDG idea is local elimination. For each element \(K\), the local linearized system can be written schematically as

\[ \begin{bmatrix} A_K & B_K \\ C_K & D_K \end{bmatrix} \begin{bmatrix} \Delta y_K \\ \Delta \hat{u}_{\partial K} \end{bmatrix} = \begin{bmatrix} r_K \\ g_K \end{bmatrix}, \]

where \(y_K\) contains element-local unknowns such as \(q_K\) and \(u_K\). The element correction is condensed as

\[ \Delta y_K = A_K^{-1}(r_K - B_K \Delta\hat{u}_{\partial K}), \]

and the global system is assembled only for trace corrections:

\[ H \Delta \hat{u} = b. \]

After the trace solve, element states are recovered locally.

Exasim HDG Implementation¶

In Exasim, HDG is the matrix-based path:

element residuals and Jacobian blocks are evaluated from generated model callbacks;
local matrices are condensed into face/trace contributions;
GMRES solves the global trace system;
preconditioners operate on trace-space blocks or Schwarz subproblems;
local element states are updated after the trace correction.

The model contract for HDG requires boundary and flux Jacobians in addition to the residual callbacks. See Model contract.

Trace Unknowns¶

The trace \(\hat{u}\) lives on faces rather than inside elements. This reduces the number of globally coupled unknowns for high-order elements because only face degrees of freedom participate in the global system.

Quantity	Location	Role
\(u_h\)	Element interior	Primary solution reconstructed locally.
\(q_h\)	Element interior	Auxiliary gradient or flux variable.
\(\hat{u}_h\)	Mesh skeleton	Globally coupled trace unknown.
Numerical flux	Element faces	Enforces interface conservation and boundary conditions.

LDG And HDG Comparison¶

Feature	LDG	HDG
Global unknowns	Element state unknowns	Trace unknowns on faces
Auxiliary variable \(q\)	Local recovery inside residual path	Local element unknown condensed against trace
Matrix assembly	Matrix-free GMRES path	Matrix-based trace-system path
Matrix-vector product	Residual differencing or derivative path	Assembled/condensed trace operator
Memory footprint	Lower matrix storage, Krylov vectors over state DOFs	Stores element/trace blocks, Krylov vectors over trace DOFs
Preconditioning	Local and Schwarz-style state-space preconditioning	Block Jacobi and additive Schwarz trace-space preconditioning
Scalability tradeoff	More global DOFs but less matrix storage	Fewer global DOFs but more local assembly/factorization work

When To Choose HDG¶

Choose HDG when:

diffusion or mixed formulations dominate the model;
reducing global unknowns is more important than avoiding matrix assembly;
trace-space preconditioning is effective for the target problem;
high polynomial orders make element-local static condensation attractive;
you need the solver path used by many production HDG examples.

Benchmark LDG and HDG for new applications. The best choice depends on polynomial order, mesh size, physics stiffness, preconditioner quality, and backend memory bandwidth.