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

A fast solver for a class of linear systems

“…It was shown in [20] that O(log n) iterations of solves produce a good approximation to a basic eigenvector of a graph. More closely related to preconditioned iterative methods is a solver for elliptic finite element linear systems [4]. This work showed that such systems can be preconditioned with graph Laplacians and so they can be solved in nearly-linear time.…”

Perturbation theory for the LDU factorization and accurate computations for diagonally dominant matrices

“…This factorization exists by part (a) of Theorem 3 applied to B [s] . Use (8), (26), part (a) of Lemma 4 and part (a) of Lemma 7 to show…”

“…For instance, they arise out of finite difference and finite element discretizations of partial differential equations and in the solution of Markov modeling problems [1,5,8,26]. Diagonally dominant matrices enjoy excellent theoretical and numerical properties that are explained in classical references [7,[29][30][31][32].…”

Hierarchical Diagonal Blocking and Precision Reduction Applied to Combinatorial Multigrid

“…Our choice is motivated by the potential for immediate impact on the design of industrial strength code for important applications. In contrast to AMG, CMG offers strong convergence guarantees for the class of symmetric diagonally dominant (SDD) matrices [23,13,4], and under certain conditions for the even more general class of symmetric M -matrices [17]. The convergence guarantees are based on recent progress in spectral graph theory and combinatorial preconditioning (see for example [12], [27]).…”