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Approximating inverse FEM matrices on non-uniform meshes with $\mathcal{H}$-matrices

Niklas Angleitner, Markus Faustmann, Jens Markus Melenk

TL;DR

The paper addresses the efficient approximation of the inverse FEM stiffness matrix for non-uniform meshes using the data-sparse $\mathcal{H}$-matrix framework. By linking matrix blocks to function-level objects through a discrete solution operator and carefully designed subspaces, discrete cut-off operators, and coarsening on locally bounded meshes, the authors prove exponential convergence in block rank and provide favorable storage bounds. The results generalize prior quasiuniform-mesh analyses to graded and locally refined meshes, with a corollary for algebraically graded meshes, and yield practical avenues for approximate direct solvers and preconditioners in iterative schemes. Numerically, the approach demonstrates exponential error decay with rank and scalable memory usage, supporting its potential for large-scale PDE solvers with repeated right-hand sides and complex geometries.

Abstract

We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse $\mathcal{H}$-matrix format. For a large class of shape regular but possibly non-uniform meshes including graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the $\mathcal{H}$-matrix format at an exponential rate in the block rank. Since the storage complexity of the hierarchical matrix is logarithmic-linear and only grows linearly in the block-rank, we obtain an efficient approximation that can be used, e.g., as an approximate direct solver or preconditioner for iterative solvers.

Approximating inverse FEM matrices on non-uniform meshes with $\mathcal{H}$-matrices

TL;DR

The paper addresses the efficient approximation of the inverse FEM stiffness matrix for non-uniform meshes using the data-sparse -matrix framework. By linking matrix blocks to function-level objects through a discrete solution operator and carefully designed subspaces, discrete cut-off operators, and coarsening on locally bounded meshes, the authors prove exponential convergence in block rank and provide favorable storage bounds. The results generalize prior quasiuniform-mesh analyses to graded and locally refined meshes, with a corollary for algebraically graded meshes, and yield practical avenues for approximate direct solvers and preconditioners in iterative schemes. Numerically, the approach demonstrates exponential error decay with rank and scalable memory usage, supporting its potential for large-scale PDE solvers with repeated right-hand sides and complex geometries.

Abstract

We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse -matrix format. For a large class of shape regular but possibly non-uniform meshes including graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the -matrix format at an exponential rate in the block rank. Since the storage complexity of the hierarchical matrix is logarithmic-linear and only grows linearly in the block-rank, we obtain an efficient approximation that can be used, e.g., as an approximate direct solver or preconditioner for iterative solvers.

Paper Structure

This paper contains 19 sections, 24 theorems, 107 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. The model problem
  4. The mesh
  5. The basis- and dual functions
  6. The system matrix
  7. Hierarchical matrices
  8. The main result
  9. Proof of main result
  10. Overview
  11. Examples of meshes with locally bounded cardinality
  12. Examples of dual functions
  13. A representation formula for the inverse system matrix
  14. Reduction from matrix level to function level
  15. The discrete cut-off operator
  16. ...and 4 more sections

Key Result

Theorem 2.13

Let $\mathcal{T} \subseteq \mathrm{Pow}(\Omega)$ be a mesh of locally bounded cardinality for some $\sigma_{\mathrm{card}} \geq 1$ in the sense of Mesh_loc_bd_card and $\{\varphi_1,\dots,\varphi_N\} \subseteq \mathbb{S}^{p,1}_{0}(\mathcal{T} )$ a basis that has a system of local dual functions (see

Figures (2)

  • Figure 1: The mesh $\mathcal{T}$, the cluster tree $\mathbb{T}_{N}^{}$ and the block partition $\mathbb{P}$ for $N \approx 2.000$ degrees of freedom.
  • Figure 2: Approximation error and memory allocation for $N \approx 72.000$ degrees of freedom.

Theorems & Definitions (65)

  • Definition 2.1
  • Definition 2.2: Mesh
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: Spline spaces
  • Definition 2.6: Basis with local dual functions
  • Remark 2.7
  • Definition 2.8: Index patches
  • Definition 2.9
  • Definition 2.10
  • ...and 55 more