# Extra Chapter 2: iHDG – Iterative Hybridized Discontinuous Galerkin method (Part 2)

This post is based on my joint work with Tan Bui-Thanh and Sriramkrishnan Muralikrishnan.

We continue our development of the previous work on iHDG method

iHDG: An Iterative HDG Framework for Partial Differential Equations. SIAM Journal on Scientific Computing.

Video 6: Our simulations for a hyperbolic system using our new iHDG

We show that the new approach has several advantages, for instance, it is well known that Domain Decomposition Methods (DDMs) encouter serious instability issues when the subdomains have cross-points or irregular shapes; moreover, the geometry of the decomposition also has a profound influence on the method. The iHDG method introduced in is indeed a two-tier parallelism: a coarse-grained parallelism on big groups of subdomains-elements, and a fine-grained parallelism on elements level within each subdomain-element. This method does not rely on any specific partition of the computational domain, and hence is independent of the geometry of the decomposition.

The second main feature of the iHDG framework is that it is provably convergent. Besides the DDM point of view, the method can also be seen from a linear algebra point of view as a block Gauss-Seidel iterative solver for the augmented HDG system with volume and trace unknowns. This point of view is not the usual way of looking at the HDG system since the linear system is assembled for trace unknowns only. For iHDG, since global matrices are never formed, this gives us the flexibility to create an efficient solver completely depending on independent element-by-element calculations. The iHDG approach is completely different from traditional Gauss-Seidel schemes for convection-diffusion problems or pure advection problems in which the unknowns are ordered in the flow direction for convergence. Robust Gauss-Seidel smoothers which depend on the ordering of the unknowns are also developed for discontinuous Galerkin methods are then developed. One drawback of this approach is that for a complex velocity field (e.g. hyperbolic systems) it is not trivial to obtain a mesh and an ordering which coincide with the flow direction. Another disadvantage of this approach is that the point or the block Gauss-Seidel scheme (for the trace system alone) requires a lot of communication between processors for calculations within an iteration. These aspects affect the scalabilty of these schemes to a large extent and in general are not favorable for parallelization. The iHDG approach is, unlike purely algebraic traditional Gauss-Seidel methods, is built upon, and hence exploiting, the HDG discretization. Of importance is the upwind flux, or more specifically the upwind stabilization, that automatically determines the flow directions. Consequently the convergence of iHDG is independent of the ordering of the unknowns.

Another crucial strong point of the iHDG method is that similar to HDG methods, each iHDG iteration consists of only independent element-by-element local solves to compute the volume unknowns. Thanks to the compact stencil of the HDG framework, this is overlapped by a single communication of the trace of the volume unknowns restricted on faces shared between the neighboring processors. Therefore, the communication requirement is thus similar to that of block Jacobi methods. The iHDG approach is then potentially adaptable to computing systems with massive concurrences.