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Solving Elliptic Finite Element Systems in Near-Linear Time with Support Preconditioners

Erik Boman, Bruce Hendrickson, Stephen Vavasis

TL;DR

The paper tackles solving FEM-based linear systems from scalar elliptic PDEs by showing the stiffness matrix $K$ can be well approximated by a symmetric diagonally dominant matrix $\bar{K}$ using support theory. It derives a factorization $K=A^T J^T D J A$ and reduces the approximation to a graph-like form $\bar{K}=A^T \bar{D} A$, with a rigorous bound on $\kappa(K,\bar{K})$ that depends only on mesh quality measures and quadrature properties, not problem size. This enables nearly linear-time solution via graph-based preconditioners (and related CG/EEST methods), with iteration counts tied to $\kappa(H)$, itself bounded by mesh and quadrature parameters. The approach is demonstrated through detailed theory, element-wise factorizations, and computational tests, and opens pathways to extensions to more general PDEs and vector problems, subject to mesh quality and problem structure. The practical impact is a scalable, theoretically grounded preconditioning framework that extends graph-theoretic methods to a broad class of FEM systems while separating mesh design from problem size.

Abstract

We consider linear systems arising from the use of the finite element method for solving scalar linear elliptic problems. Our main result is that these linear systems, which are symmetric and positive semidefinite, are well approximated by symmetric diagonally dominant matrices. Our framework for defining matrix approximation is support theory. Significant graph theoretic work has already been developed in the support framework for preconditioners in the diagonally dominant case, and in particular it is known that such systems can be solved with iterative methods in nearly linear time. Thus, our approximation result implies that these graph theoretic techniques can also solve a class of finite element problems in nearly linear time. We show that the support number bounds, which control the number of iterations in the preconditioned iterative solver, depend on mesh quality measures but not on the problem size or shape of the domain.

Solving Elliptic Finite Element Systems in Near-Linear Time with Support Preconditioners

TL;DR

The paper tackles solving FEM-based linear systems from scalar elliptic PDEs by showing the stiffness matrix can be well approximated by a symmetric diagonally dominant matrix using support theory. It derives a factorization and reduces the approximation to a graph-like form , with a rigorous bound on that depends only on mesh quality measures and quadrature properties, not problem size. This enables nearly linear-time solution via graph-based preconditioners (and related CG/EEST methods), with iteration counts tied to , itself bounded by mesh and quadrature parameters. The approach is demonstrated through detailed theory, element-wise factorizations, and computational tests, and opens pathways to extensions to more general PDEs and vector problems, subject to mesh quality and problem structure. The practical impact is a scalable, theoretically grounded preconditioning framework that extends graph-theoretic methods to a broad class of FEM systems while separating mesh design from problem size.

Abstract

We consider linear systems arising from the use of the finite element method for solving scalar linear elliptic problems. Our main result is that these linear systems, which are symmetric and positive semidefinite, are well approximated by symmetric diagonally dominant matrices. Our framework for defining matrix approximation is support theory. Significant graph theoretic work has already been developed in the support framework for preconditioners in the diagonally dominant case, and in particular it is known that such systems can be solved with iterative methods in nearly linear time. Thus, our approximation result implies that these graph theoretic techniques can also solve a class of finite element problems in nearly linear time. We show that the support number bounds, which control the number of iterations in the preconditioned iterative solver, depend on mesh quality measures but not on the problem size or shape of the domain.

Paper Structure

This paper contains 11 sections, 7 theorems, 54 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Support Theory
  3. Finite element analysis
  4. Condition numbers and assumptions
  5. The matrix approximation
  6. The first factorization
  7. The second factorization
  8. Preconditioning Strategy Summary
  9. Notes added in revision
  10. Computational tests
  11. Open questions

Key Result

lemma 1

Suppose $V\in{\mathbb{R}}^{n\times p}$, and suppose $H\in{\mathbb{R}}^{p\times p}$ is SPD. Then where $\lambda_{\max}$ denotes the largest eigenvalue.

Theorems & Definitions (13)

  • lemma 1
  • proof
  • lemma 2
  • proof
  • theorem 1
  • proof
  • lemma 3
  • proof
  • theorem 2
  • lemma 4
  • ...and 3 more