Hierarchical matrix approximability of inverse of convection dominated finite element matrices

TL;DR

The paper addresses compressing the inverse of convection-diffusion finite element matrices in the advection-dominated regime. It introduces tube-cluster partitions aligned with the convection field and establishes a Peclet-robust Caccioppoli inequality to prove low-rank approximation for block interactions, extending H-matrix theory from elliptic to convection-dominated problems. A deformation-based approach handles non-constant advection by locally straightening field lines, enabling a hierarchical LU factorization with quasi-linear complexity. Numerical experiments on unstructured grids confirm robustness with respect to the Péclet number and demonstrate practical compression and accuracy gains for both Dirichlet and Neumann problems. Overall, the method broadens the applicability of H-matrices to a broader class of PDEs by embedding physics-aware clustering into the hierarchical framework.

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 Péclet-robust Caccioppoli inequality, crucial for handling convection-dominated problems.

Figures (22)

• Figure 1: Cutoff function $\eta$ on nested domains
• Figure 2: Two splitting producing tube clusters
• Figure 3: Impact of the numbering, on the left two splitting and on the right the block matrix induced
• Figure 4: A solution and the maximal solution of \ref{['fieldline']}
• Figure 5: Tube clusters in $\Omega$ and $\bar{\Omega}$

Theorems & Definitions (14)

• Definition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• Remark 2.6
• Definition 3.1
• Remark 3.2