Domain decomposition methods

Domain decomposition methods

Domain decomposition methods for non-matching grids [1]

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 or GMRES.

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.

See also

• Multigrid method

External links

• The official Domain Decomposition Methods page
• Domain Decomposition - Numerical Simulations page

This applied mathematics-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Domain decomposition methods Applied mathematics stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем написать курсовую United States presidential election in Missouri, 1820 Hektoen enteric agar Look at other dictionaries: Balancing domain decomposition — In numerical analysis, the balancing domain decomposition method (BDD) is an iterative method to find the solution of a symmetric positive definite system of linear algebraic equations arising from the finite element method J. Mandel, Balancing… …   Wikipedia Domain — may refer to: General Territory (administrative division), a non sovereign geographic area which has come under the authority of another government Public domain, a body of works and knowledge without proprietary interest Eminent domain, the… …   Wikipedia Decomposition method — is a generic term for solutions of various problems and design of algorithms in which the basic idea is to decompose the problem into question into subproblems. The term may specifically refer to one of the following. Decomposition method… …   Wikipedia Decomposition (computer science) — Decomposition in computer science, also known as factoring, refers to the process by which a complex problem or system is broken down into parts that are easier to conceive, understand, program, and maintain. Contents 1 Overview 2 Decomposition… …   Wikipedia Decomposition method (constraint satisfaction) — In constraint satisfaction, a decomposition method translates a constraint satisfaction problem into another constraint satisfaction problem that is binary and acyclic. Decomposition methods work by grouping variables into sets, and solving a… …   Wikipedia Mortar methods — are discretization methods for partial differential equations, which use separate finite element discretization on nonoverlapping subdomains. The meshes on the subdomains do not match on the interface, and the equality of the solution is enforced …   Wikipedia Neumann–Neumann methods — In mathematics, Neumann–Neumann methods are domain decomposition preconditioners named so because they solve a Neumann problem on each subdomain on both sides of the interface between the subdomains.[1] Just like all domain decomposition methods …   Wikipedia Meshfree methods — are a particular class of numerical simulation algorithms for the simulation of physical phenomena. Traditional simulation algorithms relied on a grid or a mesh, meshfree methods in contrast use the geometry of the simulated object directly for… …   Wikipedia Neumann-Neumann methods — are domain decomposition preconditioners named so because they solve a Neumann problem on each subdomain on both sides of the interface between the subdomains. A. Klawonn and O. B. Widlund, FETI and Neumann Neumann iterative substructuring… …   Wikipedia List of mathematics-based methods — This is a list of mathematics based methods, by Wikipedia page.See also list of graphical methods.*Adams method (differential equations) *Akra Bazzi method (asymptotic analysis) *Condorcet method (voting systems) *Coombs method (voting systems)… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”