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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 inherits 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 converge...

iHDG: An Iterative HDG Framework for Partial Differential Equations