In numerical analysis, the Schur complement method, named after Issai Schur, is the basic and the earliest version of nonoverlapping domain decomposition method, also called iterative substructuring. A finite element problem is split into nonoverlapping subdomains, and the unknowns in the interiors of the subdomains are eliminated. The remaining Schur complement system on the unknowns associated with subdomain interfaces is solved by the conjugate gradient method.
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Math for Big Data, Lecture 3, Schur Decomposition

Schur Triangulation lemma / decomposition (upper trangle)

RT4.2. Schur's Lemma (Expanded)
Transcription
The method and implementation
Suppose we want to solve the Poisson equation
on some domain Ω. When we discretize this problem we get an Ndimensional linear system AU = F. The Schur complement method splits up the linear system into subproblems. To do so, divide Ω into two subdomains Ω_{1}, Ω_{2} which share an interface Γ. Let U_{1}, U_{2} and U_{Γ} be the degrees of freedom associated with each subdomain and with the interface. We can then write the linear system as
where F_{1}, F_{2} and F_{Γ} 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 U_{Γ}, 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 discrete version of the Poincaré–Steklov operator, also called the Dirichlet to Neumann mapping.
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 secondorder problems, such as the Laplace equation or linear elasticity, the matrix of the system has condition number of the order 1/h^{2}, where h is the characteristic element size. The Schur complement, however, has condition number only of the order 1/h.
For performances, the Schur complement method is combined with preconditioning, at least a diagonal preconditioner. The Neumann–Neumann method and the Neumann–Dirichlet method are the Schur complement method with particular kinds of preconditioners.