In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains. A coarse problem with one or few unknowns per subdomain is used to further coordinate the solution between the subdomains globally. The problems on the subdomains are independent, which makes domain decomposition methods suitable for parallel computing. Domain decomposition methods are typically used as preconditioners for Krylov space iterative methods, such as the conjugate gradient method, GMRES, and LOBPCG.
In overlapping domain decomposition methods, the subdomains overlap by more than the interface. Overlapping domain decomposition methods include the Schwarz alternating method and the additive Schwarz method. Many domain decomposition methods can be written and analyzed as a special case of the abstract additive Schwarz method.
In non-overlapping methods, the subdomains intersect only on their interface. In primal methods, such as Balancing domain decomposition and BDDC, the continuity of the solution across subdomain interface is enforced by representing the value of the solution on all neighboring subdomains by the same unknown. In dual methods, such as FETI, the continuity of the solution across the subdomain interface is enforced by Lagrange multipliers. The FETI-DP method is hybrid between a dual and a primal method.
Non-overlapping domain decomposition methods are also called iterative substructuring methods.
Mortar methods are discretization methods for partial differential equations, which use separate discretization on nonoverlapping subdomains. The meshes on the subdomains do not match on the interface, and the equality of the solution is enforced by Lagrange multipliers, judiciously chosen to preserve the accuracy of the solution. In the engineering practice in the finite element method, continuity of solutions between non-matching subdomains is implemented by multiple-point constraints.
Finite element simulations of moderate size models require solving linear systems with millions of unknowns. Several hours per time step is an average sequential run time, therefore, parallel computing is a necessity. Domain decomposition methods embody large potential for a parallelization of the finite element methods, and serve a basis for distributed, parallel computations.
and 23 Related for: Domain decomposition methods information
Overlapping domaindecompositionmethods include the Schwarz alternating method and the additive Schwarz method. Many domaindecompositionmethods can be written...
Overlapping domaindecompositionmethods include the Schwarz alternating method and the additive Schwarz method. Many domaindecompositionmethods can be written...
optimization Adomian decompositionmethod, a non-numerical method for solving nonlinear differential equations Domaindecompositionmethods in mathematics,...
In numerical analysis, the balancing domaindecompositionmethod (BDD) is an iterative method to find the solution of a symmetric positive definite system...
DomainDecompositionMethods - Algorithms and Theory, Springer Series in Computational Mathematics, Vol. 34, 2004 The official DomainDecomposition Methods...
Cros, A preconditioner for the Schur complement domaindecompositionmethod, in DomainDecompositionMethods in Science and Engineering, I. Herrera, D. E...
Two-level domaindecompositionmethods with Lagrange multipliers for the fast iterative solution of acoustic scattering problems, Computer Methods in Applied...
iterative domaindecompositionmethods such as FETI and balancing domaindecomposition In the engineering practice in the finite element method, continuity...
Schur complement method, named after Issai Schur, is the basic and the earliest version of non-overlapping domaindecompositionmethod, also called iterative...
the methodsDomaindecompositionmethods — divides the domain in a few subdomains and solves the PDE on these subdomains Additive Schwarz method Abstract...
multiresolution methods, very useful in problems exhibiting multiple scales of behavior. For example, many basic relaxation methods exhibit different...
without reference to domains, subdomains, etc. Many if not all domaindecompositionmethods can be cast as abstract additive Schwarz method, which is often...
\mathbb {R} ^{n})} defined on a domain V ⊆ R n {\displaystyle V\subseteq \mathbb {R} ^{n}} , a Helmholtz decomposition is a pair of vector fields G ∈ C...
whose research concerns numerical methods for partial differential equations, and particularly domaindecompositionmethods, with applications in scientific...
decomposition. One can use sensitivity indices (see variance-based sensitivity analysis) to define the most influential variables for decomposition or...
mathematics, the fictitious domainmethod is a method to find the solution of a partial differential equations on a complicated domain D {\displaystyle D} ,...
Chan, Tony F. "Introduction". Third International Symposium on DomainDecompositionMethods for Partial Differential Equations. SIAM. p. xv. ISBN 978-0-89871-253-7...
m\times n} matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an m × n {\displaystyle m\times n} complex...
in 2005. Her dissertation, A class of alternate strip-based domaindecompositionmethods for elliptic partial differential equations, was supervised by...
proper generalized decomposition is a method characterized by a variational formulation of the problem, a discretization of the domain in the style of the...
Proper Orthogonal Decomposition (POD), as it was originally formulated in the domain of fluid dynamics to analyze turbulences, is to decompose a random vector...
Global Information