# A Multilevel Approach for Trace System in HDG Discretizations

## TL;DR

This paper introduces a multilevel solver for trace systems in HDG discretizations, combining nested dissection, domain decomposition, and high-order techniques to improve efficiency and scalability across various PDEs.

## Contribution

It develops a novel two-level solver integrating coarse and fine scale methods, applicable to HDG and other high-order finite element methods, with demonstrated robustness.

## Key findings

- Performance depends only on solution smoothness, not PDE type
- Solver is robust and scalable across different PDEs
- Algorithms can be viewed as a specialized multigrid method

## Abstract

We propose a multilevel approach for trace systems resulting from hybridized discontinuous Galerkin (HDG) methods. The key is to blend ideas from nested dissection, domain decomposition, and high-order characteristic of HDG discretizations. Specifically, we first create a coarse solver by eliminating and/or limiting the front growth in nested dissection. This is accomplished by projecting the trace data into a sequence of same or high-order polynomials on a set of increasingly $h-$coarser edges/faces. We then combine the coarse solver with a block-Jacobi fine scale solver to form a two-level solver/preconditioner. Numerical experiments indicate that the performance of the resulting two-level solver/preconditioner depends only on the smoothness of the solution and is independent of the nature of the PDE under consideration. While the proposed algorithms are developed within the HDG framework, they are applicable to other hybrid(ized) high-order finite element methods. Moreover, we show that our multilevel algorithms can be interpreted as a multigrid method with specific intergrid transfer and smoothing operators. With several numerical examples from Poisson, pure transport, and convection-diffusion equations we demonstrate the robustness and scalability of the algorithms.