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An overlapping Schwarz algorithm for almost incompressible elasticity
 SIAM J. Numer. Anal
"... Abstract. Overlapping Schwarz methods are extended to mixed finite element approximations of linear elasticity which use discontinuous pressure spaces. The coarse component of the preconditioner is based on a lowdimensional space previously developed for scalar elliptic problems and a domain decomp ..."
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Cited by 25 (7 self)
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Abstract. Overlapping Schwarz methods are extended to mixed finite element approximations of linear elasticity which use discontinuous pressure spaces. The coarse component of the preconditioner is based on a lowdimensional space previously developed for scalar elliptic problems and a domain decomposition method of iterative substructuring type, i.e., a method based on nonoverlapping decompositions of the domain, while the local components of the preconditioner are based on solvers on a set of overlapping subdomains. A bound is established for the condition number of the algorithm which grows in proportion to the square of the logarithm of the number of degrees of freedom in individual subdomains and the third power of the relative overlap between the overlapping subdomains, and which is independent of the Poisson ratio as well as jumps in the Lamé parameters across the interface between the subdomains. A positive definite reformulation of the discrete problem makes the use of the standard preconditioned conjugate gradient method straightforward. Numerical results, which include a comparison with problems of compressible elasticity, illustrate the findings.
Domain decomposition for less regular subdomains: Overlapping Schwarz in two dimensions
 SIAM J. Numer. Anal
"... Abstract. In the theory of domain decomposition methods, it is often assumed that each subdomain is the union of a small set of coarse triangles or tetrahedra. In this study, extensions to the existing theory which accommodate subdomains with much less regular shapes are presented; the subdomains ar ..."
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Cited by 22 (10 self)
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Abstract. In the theory of domain decomposition methods, it is often assumed that each subdomain is the union of a small set of coarse triangles or tetrahedra. In this study, extensions to the existing theory which accommodate subdomains with much less regular shapes are presented; the subdomains are required only to be John domains. Attention is focused on overlapping Schwarz preconditioners for problems in two dimensions with a coarse space component of the preconditioner, which allows for good results even for coefficients which vary considerably. It is shown that the condition number of the domain decomposition method is bounded by C(1 + H/δ)(1 + log(H/h)) 2, where the constant C is independent of the number of subdomains and possible jumps in coefficients between subdomains. Numerical examples are provided which confirm the theory and demonstrate very good performance of the method for a variety of subregions including those obtained when a mesh partitioner is used for the domain decomposition.
Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity
 Internat. J. Numer. Meth. Engng
, 2010
"... Abstract. Overlapping Schwarz methods are considered for mixed finite element approximations of linear elasticity, with discontinuous pressure spaces, as well as for compressible elasticity approximated by standard conforming finite elements. The coarse components of the preconditioners are based on ..."
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Cited by 9 (4 self)
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Abstract. Overlapping Schwarz methods are considered for mixed finite element approximations of linear elasticity, with discontinuous pressure spaces, as well as for compressible elasticity approximated by standard conforming finite elements. The coarse components of the preconditioners are based on spaces, with a number of degrees of freedom per subdomain which is uniformly bounded, and which are similar to those previously developed for scalar elliptic problems and domain decomposition methods of iterative substructuring type, i.e., methods based on nonoverlapping decompositions of the domain. The local components of the new preconditioners are based on solvers on a set of overlapping subdomains. In the current study, the dimension of the coarse spaces is smaller than in recently developed algorithms; in the compressible case all independent face degrees of freedom have been eliminated while in the almost incompressible case five out six are not needed. In many cases, this will result in a reduction of the dimension of the coarse space by about one half compared to that of the algorithm previously considered. In spite of using overlapping subdomains to define the local components of the preconditioner, only values on the interface between the subdomains need to be retained in the iteration of the new hybrid Schwarz algorithm. The use of discontinuous pressures makes it possible to work exclusively with symmetric, positive definite problems and the standard preconditioned conjugate gradient method. Bounds are established for the condition number of the preconditioned operators. The bound for the almost incompressible case grows in proportion to the square of the logarithm of the number of degrees of freedom of individual subdomains and the third power of the relative overlap between the overlapping subdomains, and it is independent of the Poisson ratio as well as jumps in the Lamé parameters across the interface between the subdomains. Numerical results illustrate the findings.
Accomodating irregular subdomains in domain decomposition theory
 In Proceedings of the Eighteenth International Conference on Domain Decomposition Methods, number 70 in SpringerVerlag, Lecture Notes in Computational Science and Engineering
, 2009
"... Summary. In the theory for domain decomposition methods, it has previously often been assumed that each subdomain is the union of a small set of coarse shaperegular triangles or tetrahedra. Recent progress is reported, which makes it possible to analyze cases with irregular subdomains such as those ..."
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Cited by 8 (1 self)
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Summary. In the theory for domain decomposition methods, it has previously often been assumed that each subdomain is the union of a small set of coarse shaperegular triangles or tetrahedra. Recent progress is reported, which makes it possible to analyze cases with irregular subdomains such as those produced by mesh partitioners. The goal is to extend the analytic tools so that they work for problems on subdomains that might not even be Lipschitz and to characterize the rates of convergence of domain decomposition methods in terms of a few, easy to understand, geometric parameters of the subregions. For two dimensions, some best possible results have already been obtained for scalar elliptic and compressible and almost incompressible linear elasticity problems; the subdomains should be John or Jones domains and the rates of convergence are determined by parameters that characterize such domains and that of an isoperimetric inequality. Technical issues for three dimensional problems will also be discussed. 1
An iterative substructuring algorithm for twodimensional problems in H(curl
, 2010
"... Abstract. A domain decomposition algorithm, similar to classical iterative substructuring algorithms, is presented for twodimensional problems in the space H0(curl; Ω). It is defined in terms of a coarse space and local subspaces associated with individual edges of the subdomains into which the dom ..."
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Cited by 7 (4 self)
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Abstract. A domain decomposition algorithm, similar to classical iterative substructuring algorithms, is presented for twodimensional problems in the space H0(curl; Ω). It is defined in terms of a coarse space and local subspaces associated with individual edges of the subdomains into which the domain of the problem has been subdivided. The algorithm differs from others in three basic respects. First, it can be implemented in an algebraic manner that does not require access to individual subdomain matrices or a coarse discretization of the domain; this is in contrast to algorithms of the BDDC, FETI–DP, and classical two–level overlapping Schwarz families. Second, favorable condition number bounds can be established over a broader range of subdomain material properties than in previous studies. Third, we are able to develop theory for quite irregular subdomains and bounds for the condition number of our preconditioned conjugate gradient algorithm, which depend only on a few geometric parameters. The coarse space for the algorithm is based on simple energy minimization concepts, and its dimension equals the number of subdomain edges. Numerical results are presented which confirm the theory and demonstrate the usefulness of the algorithm for a variety of mesh decompositions and distributions of material properties. Key words. domain decomposition, iterative substructuring, H(curl), Maxwell’s equations, preconditioners, irregular subdomain boundaries, discontinuous coefficients
Energy Minimizing Coarse Spaces for TwoLevel Schwarz Methods for Multiscale PDEs
, 2008
"... Twolevel overlapping Schwarz methods for elliptic partial differential equations combine local solves on overlapping domains with a global solve of a coarse approximation of the original problem. To obtain robust methods for equations with highly varying coefficients, it is important to carefully c ..."
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Cited by 5 (4 self)
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Twolevel overlapping Schwarz methods for elliptic partial differential equations combine local solves on overlapping domains with a global solve of a coarse approximation of the original problem. To obtain robust methods for equations with highly varying coefficients, it is important to carefully choose the coarse approximation. Recent theoretical results by the authors have shown that bases for such robust coarse spaces should be constructed such that the energy of the basis functions is minimized. We give a simple derivation of a method that finds such a minimum energy basis using one local solve per coarse space basis function and one global solve to enforce a partition of unity constraint. Although this global solve may seem prohibitively expensive, we demonstrate that a onelevel overlapping Schwarz method is an effective and scalable preconditioner and we show that such a preconditioner can be implemented efficiently using the ShermanMorrisonWoodbury formula. The result is an elegant, scalable, algebraic method for constructing a robust coarse space given only the supports of the coarse space basis functions. Numerical experiments on a simple twodimensional model problem with a variety of binary and multiscale coefficients confirm this. Numerical experiments also show that, when used in a twolevel preconditioner, the energy minimizing coarse space gives better results than other coarse space constructions, such as the multiscale finite element approach.
Extending theory for domain decomposition algorithms to irregular subdomains
 in Proceedings of the 17th International Conference on Domain Decomposition Methods in Science and Engineering
"... In the theory of iterative substructuring domain decomposition methods, we typically assume that each subdomain is quite regular, e.g., the union of a small set of coarse triangles or tetrahedra; see, e.g., [13, Assumption 4.3]. However, this is often unrealistic especially if the subdomains result ..."
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Cited by 4 (4 self)
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In the theory of iterative substructuring domain decomposition methods, we typically assume that each subdomain is quite regular, e.g., the union of a small set of coarse triangles or tetrahedra; see, e.g., [13, Assumption 4.3]. However, this is often unrealistic especially if the subdomains result from using a mesh partitioner. The
TOPS (Terascale Optimal PDE Simulations)
"... Summary. Our work has focused on the development and analysis of domain decomposition algorithms for a variety of problems arising in continuum mechanics modeling. In particular, we have extended and analyzed FETIDP and BDDC algorithms; these iterative solvers were first introduced and studied by ..."
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Summary. Our work has focused on the development and analysis of domain decomposition algorithms for a variety of problems arising in continuum mechanics modeling. In particular, we have extended and analyzed FETIDP and BDDC algorithms; these iterative solvers were first introduced and studied by Charbel Farhat and his collaborators, see A very desirable feature of these iterative substructuring and other domain decomposition algorithms is that they respect the memory hierarchy of modern parallel and distributed computing systems, which is essential for approaching peak floating point performance. The development of improved methods, together with more powerful computer systems, is making it possible to carry out simulations in three dimensions, with quite high resolution, relatively easily. This work is supported by high quality software systems, such as Argonne's PETSc library, which facilitates code development as well as the access to a variety of parallel and distributed computer systems. The success in finding scalable and robust domain decomposition algorithms for very large number of processors and very large finite element problems is, e.g., illustrated in Our work over these five and half years has, in our opinion, helped advance the knowledge of domain decomposition methods significantly. We see these methods as providing valuable alternatives to other iterative methods, in particular, those based on multigrid. In our opinion, our accomplishments also match the goals of the TOPS project quite closely.
An Overlapping Schwarz . . .
"... Overlapping Schwarz methods form one of two major families of domain decomposition methods. We consider a twolevel overlapping Schwarz method for RaviartThomas vector fields. The coarse part of the preconditioner is based on the energyminimizing extensions and the local parts are based on tradit ..."
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Overlapping Schwarz methods form one of two major families of domain decomposition methods. We consider a twolevel overlapping Schwarz method for RaviartThomas vector fields. The coarse part of the preconditioner is based on the energyminimizing extensions and the local parts are based on traditional solvers on overlapping subdomains. We show that the condition number grows linearly with the logarithm of the number of degrees of freedom in the individual subdomains and linearly with the relative overlap between the overlapping subdomains. The condition number of the method is also independent of the values and jumps of the coefficients. Numerical results for 2D and 3D problems, which support the theory, are also presented.