The balancing domain decomposition is a Neumann–Neumann method with a special kind of coarse problem.
More specifically, consider a domain Ω, on which we wish to solve the Poisson equation for some function f. Split the domain into two non-overlapping subdomains Ω1 and Ω2 with common boundary Γ and let u1 and u2 be the values of u in each subdomain.
An iterative method with iterations k=0,1,... for the approximation of each ui (i=1,2) that satisfies the matching conditions is to first solve the Dirichlet problems for some function λ(k) on Γ, where λ(0) is any inexpensive initial guess.
We then solve the two Neumann problems We then obtain the next iterate by setting for some parameters ω, θ1 and θ2.
[2] This continuous iteration can be discretized by the finite element method and then solved—in parallel—on a computer.