  We present a scalable iterative solver for high-order hybridized
discontinuous Galerkin (HDG) discretizations of linear partial differential
equations. It is an interplay between domain decomposition methods and HDG
discretizations, and hence inheriting advances from both sides. In particular,
the method can be viewed as a Gauss-Seidel approach that requires only
independent element-by-element and face-by-face local solves in each iteration.
As such, it is well-suited for current and future computing systems with
massive concurrencies. Unlike conventional Gauss-Seidel schemes which are
purely algebraic, the convergence of iHDG, thanks to the built-in HDG numerical
flux, does not depend on the ordering of unknowns. We rigorously show the
convergence of the proposed method for the transport equation, the linearized
shallow water equation and the convection-diffusion equation. For the transport
equation, the method is convergent regardless of mesh size $h$ and solution
order $p$, and furthermore the convergence rate is independent of the solution
order. For the linearized shallow water and the convection-diffusion equations
we show that the convergence is conditional on both $h$ and $p$. Extensive
steady and time-dependent numerical results for the 2D and 3D transport
equations, the linearized shallow water equation, and the convection-diffusion
equation are presented to verify the theoretical findings.
