An Improved Iterative HDG Approach for Partial Differential Equations

TL;DR

This paper introduces an improved iterative hybridized discontinuous Galerkin method for linear PDEs, achieving finite convergence, unconditional stability, and scalable performance, with insights into iteration counts relative to problem parameters.

Contribution

The paper advances the iHDG method by proving finite and unconditional convergence, enhancing stability, and analyzing iteration dependence on problem parameters, enabling better scalability and efficiency.

Findings

Finite convergence for scalar transport equation.

Unconditional convergence for shallow water and convection-diffusion equations.

Scalability demonstrated up to 16,384 cores.

Abstract

We propose and analyze an iterative high-order hybridized discontinuous Galerkin (iHDG) discretization for linear partial differential equations. We improve our previous work (SIAM J. Sci. Comput. Vol. 39, No. 5, pp. S782--S808) in several directions: 1) the improved iHDG approach converges in a finite number of iterations for the scalar transport equation; 2) it is unconditionally convergent for both the linearized shallow water system and the convection-diffusion equation; 3) it has improved stability and convergence rates; 4) we uncover a relationship between the number of iterations and time stepsize, solution order, meshsize and the equation parameters. This allows us to choose the time stepsize such that the number of iterations is approximately independent of the solution order and the meshsize; and 5) we provide both strong and weak scalings of the improved iHDG approach up to $16,384$ cores. A connection between iHDG and time integration methods such as parareal and implicit/explicit methods are discussed. Extensive numerical results are presented to verify the theoretical findings.