The construction of preconditioners for elliptic problems by substructuring. I

Abstract

Dedicated to ProfessorJoachim Nitsche on the occasion of the sixtieth anniversary of his birthday. Abstract.We consider the problem of solving the algebraic system of equations which arise from the discretization of symmetric elliptic boundary value problems via finite element methods. A new class of preconditioners for these discrete systems is developed based on substructuring (also known as domain decomposition). The resulting preconditioned algorithms are well suited to emerging parallel computing architectures.The proposed methods are applicable to problems on general domains involving differential operators with rather general coefficients. A basic theory for the analysis of the condition number of the preconditioned system (which determines the iterative convergence rate of the algorithm) is given. Techniques for applying the theory and algorithms to problems with irregular geometry are discussed and the results of extensive numerical experiments are reported.1. Introduction. The aim of this series of papers is to propose and analyze methods for efficiently solving the equations resulting from finite element discretizations of second-order elliptic boundary value problems on general domains in R2 and R3. In particular, we shall be concerned with constructing easily invertible and "effective" preconditioners for the resulting system of discrete equations which can be used in a preconditioned iterative algorithm to achieve a rapid solution method. The methods to be presented are well suited to parallel computing architectures.In this paper we shall restrict ourselves to boundary value problems in R2. Let fî be a bounded domain in R2 with a piecewise smooth boundary dû. As a model problem for a second-order uniformly elliptic equation we shall consider the

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“…There is numerical evidence (cf. [3]) that the estimate (2.14) is sharp. A mathematical proof will be given in Section 4.…”

“…In the original BPS algorithm (cf. [3]) the exact solves S −1 j are replaced by spectrally equivalent interface preconditioners that are easier to compute. But for our purpose we may as well use exact solves.…”

“…The proofs of these facts can be found either in [3], [7], or by straightforward calculations using (1.10)-(1.12).…”

“…In this paper we will establish lower bounds for two wellknown nonoverlapping domain decomposition preconditioners in two dimensions: the substructuring preconditioner of Bramble, Pasciak and Schatz (cf. [3]) and the Neumann-Neumann preconditioner (cf. [9], [10], [16], [11] and the references therein).…”

A multiscale preconditioner for stochastic mortar mixed finite elements

“…This approach is closely related (but more general) than the substructuring methods developed in e.g., [8,7,1]. If the basis functions for M H,i are chosen to be the Lagrange basis, then this procedure is similar to the construction of the local contribution to the Schur complement [39,36].…”