Hierarchical matrix approximability of inverse of convection dominated finite element matrices

TL;DR

This paper introduces a new hierarchical matrix approximation method for the inverse of convection-dominated finite element matrices, overcoming previous limitations on unstructured grids and variable convection fields.

Contribution

It proposes a convection tube-based partitioning strategy that is robust for general grids and convection fields, supported by theoretical analysis and numerical validation.

Findings

Efficient approximation of inverse matrices in convection-dominated problems.

Robustness of the method on unstructured grids.

Theoretical guarantees based on Péclet-robust inequalities.

Abstract

Several researchers have developed a rich toolbox of matrix compression techniques that exploit structure and redundancy in large matrices. Classical methods such as the block low-rank format and the Fast Multipole Method make it possible to manipulate very large systems by representing them in a reduced form. Among the most sophisticated tools in this area are hierarchical matrices (H-matrices), which exploit local properties of the underlying kernel or operator to approximate matrix blocks by low-rank factors, organized in a recursive hierarchy. H-matrices offer a flexible and scalable framework, yielding nearly linear complexity in both storage and computation. Hierarchical matrix techniques, originally developed for boundary integral equations, have recently been applied to matrices stemming from the discretization of advection-dominated problems. However, their effectiveness is limited by the loss of coercivity induced by convection phenomena, where traditional methods fail. Initial work by Le Borne addressed this by modifying the admissibility criterion for structured grids with constant convection, but challenges remain for more general grids and advection fields. In this work, we propose a novel partitioning strategy based on "convection tubes", clusters aligned with the convection vector field. This method does not require a structured grid or constant convection, overcoming the limitations of previous approaches. We present both theoretical analyses and numerical experiments, that demonstrate the efficiency and robustness of our method for convection-dominated PDEs on unstructured grids. The approach builds on a P\'eclet-robust Caccioppoli inequality, crucial for handling convection-dominated problems.