iHDG: An Iterative HDG Framework for Partial Differential Equations

TL;DR

The paper introduces iHDG, a scalable iterative solver for high-order HDG discretizations of PDEs, combining domain decomposition and HDG techniques, with proven convergence and efficiency demonstrated through extensive numerical tests.

Contribution

The paper proposes a novel iterative HDG framework that is scalable, convergent for various PDEs, and suitable for massively parallel computing systems.

Findings

Converges independently of mesh size and solution order for the transport equation.

Convergence depends on mesh size and solution order for shallow water and convection-diffusion equations.

Numerical results confirm theoretical convergence and efficiency.

Abstract

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.