# Domain decomposition methods for partial differential equations

Sunday, November 28, 2010 - 2:00pm - 3:30pm

Keller 3-180

David Keyes (King Abdullah University of Science & Technology)

Domain decomposition, a form of divide-and-conquer for mathematical problems

posed over a physical domain is the most common paradigm for large-scale

simulation on massively parallel, distributed, hierarchical memory

computers. In domain decomposition, a large problem is reduced to a

collection of smaller problems, each of which is easier to solve

computationally than the undecomposed problem, and most or all of which can

be solved independently and concurrently. Domain decomposition has proved to

be an ideal paradigm not only for execution on advanced architecture

computers, but also for the development of reusable, portable software. The

most complex operation in a typical domain decomposition method – the

application of the preconditioner – carries out in each subdomain steps

nearly identical to those required to apply a conventional preconditioner to

the undecomposed domain. Hence software developed for the global problem can

readily be adapted to the local problem, instantly presenting lots of legacy

scientific code for to be harvested for parallel implementations. Finally,

it should be noted that domain decomposition is often a natural paradigm for

the modeling community. Physical systems are often decomposed into two or

more contiguous subdomains based on phenomenological considerations,

and the subdomains are discretized accordingly, as independent

tasks. This physically-based domain decomposition may be mirrored in the

software engineering of the corresponding code, and leads to threads of

execution that operate on contiguous subdomain blocks. This tutorial

provides an overview of domain decomposition and focuses on the mathematical

development of its two main paradigms: Schwarz and Schur preconditioning and

their hybrids.

posed over a physical domain is the most common paradigm for large-scale

simulation on massively parallel, distributed, hierarchical memory

computers. In domain decomposition, a large problem is reduced to a

collection of smaller problems, each of which is easier to solve

computationally than the undecomposed problem, and most or all of which can

be solved independently and concurrently. Domain decomposition has proved to

be an ideal paradigm not only for execution on advanced architecture

computers, but also for the development of reusable, portable software. The

most complex operation in a typical domain decomposition method – the

application of the preconditioner – carries out in each subdomain steps

nearly identical to those required to apply a conventional preconditioner to

the undecomposed domain. Hence software developed for the global problem can

readily be adapted to the local problem, instantly presenting lots of legacy

scientific code for to be harvested for parallel implementations. Finally,

it should be noted that domain decomposition is often a natural paradigm for

the modeling community. Physical systems are often decomposed into two or

more contiguous subdomains based on phenomenological considerations,

and the subdomains are discretized accordingly, as independent

tasks. This physically-based domain decomposition may be mirrored in the

software engineering of the corresponding code, and leads to threads of

execution that operate on contiguous subdomain blocks. This tutorial

provides an overview of domain decomposition and focuses on the mathematical

development of its two main paradigms: Schwarz and Schur preconditioning and

their hybrids.

MSC Code:

65M55

Keywords: