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Peaceman—Rachford procedure and domain decomposition for finite element problems
Author(s) -
Layton William J.,
Rabier Patrick J.
Publication year - 1995
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1680020405
Subject(s) - domain decomposition methods , finite element method , decomposition , mathematics , domain (mathematical analysis) , element (criminal law) , mortar methods , algebra over a field , mathematical optimization , computer science , mathematical analysis , pure mathematics , structural engineering , engineering , chemistry , organic chemistry , political science , law
This paper presents a general method to associate the operator splitting for the Peaceman—Rachford procedure with a decomposition of the domain in problems arising from finite element discretization of partial differential equations. The algorithm is provably convergent without any symmetry requirement. Moreover, this method possesses the significant advantage of making the linear systems of the Peaceman—Rachford iteration block diagonal and therefore perfectly appropriate for parallel processing. Not only is sparsity not affected but a reduction of the bandwidth occurs. In fact, for appropriate choices of nonconforming finite element spaces, this method makes directly possible elementwise processing. This option remains available in general for higher‐dimensional problems by applying the splitting algorithm recursively. Practical implementation requires nothing more than the standard finite element assembly procedure and some bookkeeping to relate a few different orderings of the nodes. In addition to all these attractive features, the method is rapidly convergent and remains highly competitive even when used on a serial machine.