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Domain Decomposition Algorithms Acta Numerica pp Domain decomp osition algorithms Tony F Chan Department of Mathematics University of California at Los Angeles Los Angeles CA USA Email chanmathuclaedu Tarek P Mathew Department of Mathematics University of Wyoming Laramie WY USA Email mathewcorraluwyoedu Domain decomp osition refers to divide and conquer techniques for solving partial dierential equations by iteratively solving subproblems dened on smaller sub domains The principal advantages include enhancement of par allelism and lo calized treatment of complex and irregular geometries sin gularities and anomalous regions Additionally domain decomp osition can sometimes reduce the computational complexity of the underlying solution metho d In this article we survey iterative domain decomp osition techniques that have b een developed in recent years for solving several kinds of partial dif ferential equations including elliptic parab olic and dierential systems such as the Stokes problem and mixed formulations of elliptic problems We fo cus on describing the salient features of the algorithms and describ e them using easy to understand matrix notation In the case of elliptic problems we also provide an introduction to the convergence theory which requires some knowledge of nite element spaces and elementary functional analysis The authors were supp orted in part by the National Science Foundation under grant ASC by the Army Research Oce under contract DAALG and sub contract under DAALC and by the Oce for Naval Research under contract ONR NJ TF Chan and TP Mathew CONTENTS Introduction Overlapping sub domain algorithms Nonoverlapping sub domain algorithms Introduction to the convergence theory Some practical implementation issues Multilevel algorithms Algorithms for lo cally rened grids Domain imbedding or ctitious domain metho ds Convectiondiusion problems Parabolic problems Mixed nite elements and the Stokes problem Other topics References Introduction Domain decomp osition DD metho ds are techniques for solving partial dif ferential equations based on a decomp osition of the spatial domain of the problem into several sub domains Such reformulations are usually motivated by the need to create solvers which are easily parallelized on coarse grain parallel computers though sometimes they can also reduce the complexity of solvers on sequential computers These techniques can often b e applied directly to the partial dierential equations but they are of most interest when applied to discretizations of the dierential equations either by nite dierence nite element sp ectral or sp ectral element metho ds The primary technique consists of solving subproblems on various sub domains while enforcing suitable continuity requirements b etween adjacent subprob lems till the lo cal solutions converge within a sp ecied accuracy to the true solution In this article we fo cus on describing iterative domain decomp osition algorithms particularly on the formulation of preconditioners for solution by conjugate gradient type metho ds Though many fast direct domain de comp osition solvers have b een developed in the engineering literature see Kron and Przemieniecki these are often called substructur ing or tearing metho ds the more recent developments have b een based on the iterative approach which is p otentially more ecient in b oth time and storage The earliest known iterative domain decomp osition technique was prop osed in the pioneering work of H A Schwarz in to prove the existence of harmonic functions on irregular regions which are the union of overlapping subregions Variants of Schwarzs metho d were later studied by Sob olev Morgenstern and Babuska See also Courant Domain decomposition survey and Hilb ert The recent interest in domain decomp osition was initi ated in studies by Dinh Glowinski and Periaux Dryja Golub and Mayers Bramble Pasciak and Schatz b Bjrstad and Widlund Lions Agoshkov and Leb edev and Marchuk Kuznetsov and Matsokin where the primary motivation was the in herent parallelism of these metho ds There are not many general references that provide an overview of the eld but here are a few discussions in Keyes and Gropp Canuto Hussaini Quarteroni and Zang Xu a Dryja and Widlund Hackbusch Le Tallec and the b o oks of Leb edev Kang and Lu Shih and Liem and the forthcoming b o ok by Smith Bjrstad and Gropp The b est source of references remains the collection of conference pro ceedings Glowinski Golub Meurant and Periaux Chan Glowinski Periaux and Widlund Glowinski Kuznetsov Meurant Periaux and Widlund Chan Keyes Meurant Scroggs and Voigt a Quar teroni This article is conceptually organized in three parts The rst part Sec tions through deals with secondorder selfadjoint elliptic problems The algorithms and theory are most mature for this class of problem and the topics here are treated in more depth than in the rest of the article Most domain decomp osition metho ds can b e classied as either an overlapping or a nonoverlapping sub domain approach which we shall discuss in Sections and resp ectively A basic theoretical framework for studying the conver gence rates will b e summarized in Section Some practical implementation issues will b e discussed in Section The second part Sections consid ers algorithms that are not strictly sp eaking domain decomp osition meth o ds but that can b e studied by the general framework set up in the rst part The key idea here is to extend the concept of the sub domains to that of subspaces The topics include multilevel preconditioners Section lo cally rened grids Section and ctitious domain metho ds Section In the last part Sections we consider domain decomp osition metho ds for more general problems including convectiondiusion problems Sec tion parab olic problems Section mixed nite element metho ds and the Stokes problems Section In Section we provide references to algorithms for the biharmonic problem sp ectral element metho ds indenite problems and nonconforming nite element metho ds Due to space limita tion and the fact that b oth the theory and algorithms are generally less well developed for these problems we do not treat Parts I I and III in as much depth as in Part I Our aim is instead to highlight some of the key ideas using the framework and terminology developed in Part I and to provide a guide to the vast developing literature We present the metho ds in algorithmic form expressed in matrix notation in the hop e of making the article accessible to a broad sp ectrum of readers TF Chan and TP Mathew Given the space limitation most of the theorems esp ecially those in Parts II and I I I are stated without pro ofs with p ointers to the literature given instead We also do not cover nonlinear problems or sp ecic applications eg CFD of domain decomp osition algorithms In the rest of this section we introduce the main features of domain de comp osition pro cedures by describing several algorithms based on the sim pler case of two subdomain decomp osition for solving the following general secondorder selfadjoint co ercive elliptic problem Lu r ax y ru f x y in u on We are particularly interested in the solution of its discretization by either nite elements or nite dierences which yields a large sparse symmetric p ositive denite linear system Au f Overlapping subdomain approach Overlapping domain decomp osition algorithms are based on a decomp osition of the domain into a number of overlapping subregions Here we consider the case of two overlapping subregions f g which form a covering of see Figure We shall let i denote the part of the b oundary of i i which is in the interior of The basic Schwarz alternating algorithm to solve starts with any suitable initial guess u and constructs a sequence of improved approxima k tions u u Starting with the k th iterate u we solve the following two subproblems on and successively with the most current values as b oundary condition on the articial interior b oundaries k f on Lu k k on u j u k on n u and k f on Lu k k on u u j k on n u k The iterate u is then dened by k u x y if x y k u x y k u x y if x y n It can b e shown that in the norm induced by the op erator L the iterates k fu g converge geometrically to the true solution u on ie k k ku u k ku u k Domain decomposition survey Nonoverlapping sub domains Overlapping sub domains @ 2 1 2 1 @ @ B 1 2 @ @ @ @ 1 2 1 2 1 2 Fig Two sub domain decomp ositions where dep ends on the choice of and The ab ove Schwarz pro cedure extends almost verbatim to discretizations of We shall describ e the discrete algorithm in matrix notation Cor resp onding to the subregions f g let fI I g denote the indices of the no des in the interior of domain and interior of resp ectively Thus I and I form an overlapping set of indices for the unknown vector u Let n b e the number of indices in I and let n b e the number of indices in I Due to overlapn n n where n is the number of unknowns in Corresp onding to each region we dene a rectangular n n extension i i T matrix R whose action extends by zero a vector of no dal values in i i Thus given a subvector x of length n with no dal values at the interior i i no des on we dene i x for k I i k i T R x i k i for k I I where I I I i T The entries of the matrix R are ones or zeros The transp ose R
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