Extra Chapter 1: iHDG – Iterative Hybridized Discontinuous Galerkin method

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

Congratulations, Prof Tan Bui-Thanh for the great achievement!

Although the blog is about Quantum Kinetic Theory, I would like to, sometimes, take advantage of it to introduce some other aspects of my work. In this extra chapter, I will discuss my joint work with Tan Bui-Thanh and Sriramkrishnan Muralikrishnan, at the CEO group, Department of Aerospace Engineering and Engineering Mechanics, UT Austin. In this work, we introduce a new iterative HDG method, for computing systems with massive concurrences.

Video 1: Our simulations for contaminant transport using our new iHDG

The discontinuous Galerkin (DG) method combines advantages of classical finite volume and finite element methods. In particular, it has the ability to treat solutions with large gradients including shocks, it provides the flexibility to deal with complex geometries, and it is highly parallelizable due to its compact stencil. However, for steady state problems or time-dependent ones that require implicit time-integrators, DG methods typically have many more (coupled) unknowns compared to the other existing numerical methods, and hence more expensive in general. In order to mitigate the computational expense associated with DG methods, Cockburn, coauthors, and others have introduced the so-called hybridizable (also known as hybridized) discontinuous Galerkin (HDG) methods. In HDG discretizations, the coupled unknowns are single-valued traces introduced on the mesh skeleton, i.e. the faces, and the resulting matrix is substantially smaller and sparser compared to standard DG approaches. Once they are solved for, the usual DG unknowns can be recovered in an element-by-element fashion, completely independent of each other. Nevertheless, the trace system is still a bottleneck for practically large-scale applications, where complex and high-fidelity simulations involving features with a large range of spatial and temporal scales are necessary.

In this work, we introduce a new iterative, parallel method based on HDG solvers, to treat large-scale applications, with complex and high-fidelity simulations.

Video 2: Our simulation for linearized shallow water using our new iHDG

Video 3: Our simulation for linearized shallow water using our new iHDG

We design the iterative HDG (iHDG) method to have a two-tier parallelism to adapt to current and future computing systems: a coarse-grained parallelism on subdomains, and a fine-grained parallelism on elements level within each subdomain. In particular, the method is a fixed-point approach that requires only independent element-by-element local solves in each iteration. We view the iHDG method as an extreme Domain Decomposition Method (DDM) approach in which each subdomain is an element. For current DDMs , decomposing the computational domain into smaller subdomains encounters difficulties when the subdomains have cross-points or irregular shapes; moreover, the geometry of the decomposition also has a profound influence on the method. Unlike existing approaches, our method does not rely on any specific partition of the computational domain, and hence is independent of the geometry of the decomposition.

Video 4: Our simulation for an elliptic problem using our new iHDG

We rigorously show the convergence of the proposed method for transport equation, linearized shallow water equation and convection-diffusion equation. For 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 linearized shallow water and convection-diffusion equations we show that the convergence is conditional on both $h$ and $p$. Extensive 2D and 3D numerical results for steady and time-dependent problems are presented to verify the theoretical findings.

Video 5: Our simulation for transport using our new iHDG