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Exponential meshes and $\mathcal{H}$-matrices

Niklas Angleitner, Markus Faustmann, Jens Markus Melenk

TL;DR

This work extends exponential-in-block-rank approximability results for inverses of FEM stiffness matrices to a broader class of strongly graded meshes, including exponentially graded meshes, while making the dependence on the polynomial degree $p$ explicit. The authors develop polynomial-preserving liftings and projections, refine the cut-off and Caccioppoli inequality arguments, and replace previous delta-based constraints with an axis-parallel-box framework to achieve p-explicit, mesh-graded-invariant bounds. They prove that there exists an ${\mathcal{H}}$-matrix approximation $\boldsymbol{B}$ to $\boldsymbol{A}^{-1}$ with an error bound $\|\boldsymbol{A}^{-1}-\boldsymbol{B}\|_{2}$ that decays like $\exp(-\sigma_{\mathrm{exp}} r^{1/(d+1)} p^{-\sigma_{\mathrm{red}}})$, up to factors depending on mesh parameters and dimension. The results enable scalable, data-sparse solutions of high-order FEM systems on complex graded meshes and are supported by numerical experiments showing rapid convergence in practice.

Abstract

In our previous works, we proved that the inverse of the stiffness matrix of an $h$-version finite element method (FEM) applied to scalar second order elliptic boundary value problems can be approximated at an exponential rate in the block rank by $\mathcal{H}$-matrices. Here, we improve on this result in multiple ways: (1) The class of meshes is significantly enlarged and includes certain exponentially graded meshes. (2) The dependence on the polynomial degree $p$ of the discrete ansatz space is made explicit in our analysis. (3) The bound for the approximation error is sharpened, and (4) the proof is simplified.

Exponential meshes and $\mathcal{H}$-matrices

TL;DR

This work extends exponential-in-block-rank approximability results for inverses of FEM stiffness matrices to a broader class of strongly graded meshes, including exponentially graded meshes, while making the dependence on the polynomial degree explicit. The authors develop polynomial-preserving liftings and projections, refine the cut-off and Caccioppoli inequality arguments, and replace previous delta-based constraints with an axis-parallel-box framework to achieve p-explicit, mesh-graded-invariant bounds. They prove that there exists an -matrix approximation to with an error bound that decays like , up to factors depending on mesh parameters and dimension. The results enable scalable, data-sparse solutions of high-order FEM systems on complex graded meshes and are supported by numerical experiments showing rapid convergence in practice.

Abstract

In our previous works, we proved that the inverse of the stiffness matrix of an -version finite element method (FEM) applied to scalar second order elliptic boundary value problems can be approximated at an exponential rate in the block rank by -matrices. Here, we improve on this result in multiple ways: (1) The class of meshes is significantly enlarged and includes certain exponentially graded meshes. (2) The dependence on the polynomial degree of the discrete ansatz space is made explicit in our analysis. (3) The bound for the approximation error is sharpened, and (4) the proof is simplified.
Paper Structure (17 sections, 17 theorems, 89 equations, 4 figures)

This paper contains 17 sections, 17 theorems, 89 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. The model problem
  4. The spline spaces
  5. The system matrix
  6. Hierarchical matrices
  7. The main result
  8. Proof of main result
  9. Overview
  10. Reduction from matrix level to function space level
  11. The cut-off operator
  12. The spaces of discrete and harmonic functions
  13. The low-rank approximation operator
  14. The coarsening operators
  15. Putting everything together
  16. ...and 2 more sections

Key Result

Theorem 2.13

Let $a(\cdot,\cdot)$ be the elliptic bilinear form from Bilinear_form, let $\mathcal{T} \subseteq \mathrm{Pow}(\Omega)$ be a mesh as in Mesh, and let $p \geq 1$ be an arbitrary integer. Let $\{\varphi_1,\dots,\varphi_N\} \subseteq \mathbb{S}^{p,1}_{0}(\mathcal{T} )$ be a basis that allows for a sys

Figures (4)

  • Figure 1: From left to right: Uniform, algebraically graded towards edge, exponentially graded towards edge, exponentially graded towards corner. \ref{['Main_result_corollary']} covers the first, second and third type, but not the last one.
  • Figure 2: The lifting operator in the case $d=3$.
  • Figure 3: Left: The mesh $\mathcal{T}$. Center: The block partition $\mathbb{P}$. Right: Empirical approximation errors.
  • Figure 4: Exponential convergence of $\mathcal{H}$-matrix approximations for $p=5,6,7$.

Theorems & Definitions (52)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • Definition 2.11
  • ...and 42 more