Schwarz alternating method
Encyclopedia
In mathematics
, the Schwarz alternating method, named after Hermann Schwarz
, is an iterative method
to find the solution of a partial differential equation
s on a domain
which is the union of two overlapping subdomains, by solving the equation on each of the two subdomains in turn, taking always the latest values of the approximate solution as the boundary conditions. A modification of the method, known as the additive Schwarz method, has become a practical domain decomposition method
. An abstract formulation of the original method is then referred to as the multiplicative Schwarz method.
and served as a theoretical tool: his convergence for general second order elliptic partial differential equations was first proved much later, in 1951, by Solomon Mikhlin
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Schwarz alternating method, named after Hermann Schwarz
Hermann Schwarz
Karl Hermann Amandus Schwarz was a German mathematician, known for his work in complex analysis. He was born in Hermsdorf, Silesia and died in Berlin...
, is an iterative method
Iterative method
In computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method...
to find the solution of a partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s on a domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
which is the union of two overlapping subdomains, by solving the equation on each of the two subdomains in turn, taking always the latest values of the approximate solution as the boundary conditions. A modification of the method, known as the additive Schwarz method, has become a practical domain decomposition method
Domain decomposition 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.- Overview :...
. An abstract formulation of the original method is then referred to as the multiplicative Schwarz method.
Historical notice
It was first formulated by H. A. SchwarzHermann Schwarz
Karl Hermann Amandus Schwarz was a German mathematician, known for his work in complex analysis. He was born in Hermsdorf, Silesia and died in Berlin...
and served as a theoretical tool: his convergence for general second order elliptic partial differential equations was first proved much later, in 1951, by Solomon Mikhlin
Solomon Mikhlin
Solomon Grigor'evich Mikhlin was a Soviet mathematician of Jewish origin, working in the fields of linear elasticity, singular integrals and numerical analysis: he is best known for the introduction of the concept of "symbol of a singular integral operator", which eventually led to...
.