Schur complement method
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In
The method and implementation
Suppose we want to solve the Poisson equation
on some domain Ω. When we discretize this problem we get an N-dimensional linear system AU = F. The Schur complement method splits up the linear system into sub-problems. To do so, divide Ω into two subdomains Ω1, Ω2 which share an interface Γ. Let U1, U2 and UΓ be the degrees of freedom associated with each subdomain and with the interface. We can then write the linear system as
where F1, F2 and FΓ are the components of the load vector in each region.
The Schur complement method proceeds by noting that we can find the values on the interface by solving the smaller system
for the interface values UΓ, where we define the Schur complement matrix
The important thing to note is that the computation of any quantities involving or involves solving decoupled Dirichlet problems on each domain, and these can be done in parallel. Consequently, we need not store the Schur complement matrix explicitly; it is sufficient to know how to multiply a vector by it.
Once we know the values on the interface, we can find the interior values using the two relations
which can both be done in parallel.
The multiplication of a vector by the Schur complement is a
Advantages
There are two benefits of this method. First, the elimination of the interior unknowns on the subdomains, that is the solution of the Dirichlet problems, can be done in parallel. Second, passing to the Schur complement reduces condition number and thus tends to decrease the number of iterations. For second-order problems, such as the
For performances, the Schur complement method is combined with preconditioning, at least a
When a fast function is utilized, especially in low cost parallel computers, the Schur complement method is relatively efficient.[1]
References
- ^ Soria Guerrero, M. Schur complement method (PDF). p. 150. Retrieved 14 February 2024.