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

TL;DR

This paper demonstrates that finite element systems for elliptic problems can be efficiently approximated and solved in nearly linear time using support preconditioners, with performance depending on mesh quality.

Contribution

It introduces a novel approximation framework for finite element systems using support theory, enabling nearly linear time solutions.

Findings

Finite element systems are well approximated by diagonally dominant matrices.

Support number bounds depend on mesh quality, not problem size.

Finite element problems can be solved efficiently with graph-theoretic preconditioners.

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.