![Ddm original logo](https://upload.wikimedia.org/wikipedia/commons/a/a9/Ddm_original_logo.png)
In
mathematics,
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
, and
numerical partial differential equations
Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs).
In principle, specialized methods for hyperbolic, parabolic or elliptic parti ...
, 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 : ''This article deals with a component of numerical methods. For coarse space in topology, see coarse structure.''
In numerical analysis, coarse problem is an auxiliary system of equations used in an iterative method for the solution of a given la ...
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
preconditioner
In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for Numerical mathematics, numerical solving methods. Preconditioning is typical ...
s for
Krylov space iterative method
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the pr ...
s, such as the
conjugate gradient method,
GMRES In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of an indefinite nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace wit ...
, and
LOBPCG Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is a matrix-free method for finding the largest (or smallest) eigenvalues and the corresponding eigenvectors of a symmetric generalized eigenvalue problem
:A x= \lambda B x,
for a ...
.
In overlapping domain decomposition methods, the subdomains overlap by more than the interface. Overlapping domain decomposition methods include the
Schwarz alternating method
Schwarz may refer to:
* Schwarz, Germany, a municipality in Mecklenburg-Vorpommern, Germany
* Schwarz (surname), a surname (and list of people with the surname)
* Schwarz (musician), American DJ and producer
* Schwarz (Böhse Onkelz album), ''Schw ...
and the
additive Schwarz method In mathematics, the additive Schwarz method, named after Hermann Schwarz, solves a boundary value problem for a partial differential equation approximately by splitting it into boundary value problems on smaller domains and adding the results.
...
. Many domain decomposition methods can be written and analyzed as a special case of the
abstract additive Schwarz method
Abstract may refer to:
* ''Abstract'' (album), 1962 album by Joe Harriott
* Abstract of title a summary of the documents affecting title to parcel of land
* Abstract (law), a summary of a legal document
* Abstract (summary), in academic publish ...
.
In non-overlapping methods, the subdomains intersect only on their interface. In primal methods, such as
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 d ...
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 multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied e ...
s. The
FETI-DP The FETI-DP method is a domain decomposition methodC. Farhat, M. Lesoinne, P. LeTallec, K. Pierson, and D. Rixen, ''FETI-DP: a dual-primal unified FETI method. I. A faster alternative to the two-level FETI method'', Internat. J. Numer. Methods Engrg ...
method is hybrid between a dual and a primal method.
Non-overlapping domain decomposition methods are also called iterative substructuring methods.
Mortar method In numerical analysis, 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 ...
s 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.
Example 1: 1D Linear BVP
The exact solution is:
Subdivide the domain into two subdomains, one from