Neumann–Neumann methods

In mathematics, Neumann–Neumann methods are domain decomposition preconditioners named so because they solve a Neumann problem on each subdomain on both sides of the interface between the subdomains.^[1] Just like all domain decomposition methods, so that the number of iterations does not grow with the number of subdomains, Neumann–Neumann methods require the solution of a coarse problem to provide global communication. 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

$${\displaystyle -\Delta u=f,\qquad u|_{\partial \Omega }=0}$${\displaystyle -\Delta u=f,\qquad u|_{\partial \Omega }=0}

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. At the interface between the two subdomains, the two solutions must satisfy the matching conditions

$${\displaystyle u_{1}=u_{2},\qquad \partial _{n_{1}}u_{1}=\partial _{n_{2}}u_{2}}$${\displaystyle u_{1}=u_{2},\qquad \partial _{n_{1}}u_{1}=\partial _{n_{2}}u_{2}}

where ${\textstyle n_{i}}${\textstyle n_{i}} is the unit normal vector to Γ in each subdomain.

An iterative method with iterations k = 0, 1, ... for the approximation of each u_i (i = 1, 2) that satisfies the matching conditions is to first solve the Dirichlet problems

$${\displaystyle {\begin{aligned}-&\Delta u_{i}^{(k)}=f_{i}~~{\text{in}}~~\Omega _{i},\\[1.3ex]&\left.u_{i}^{(k)}\right|_{\partial \Omega }=0,\quad \left.u_{i}^{(k)}\right|_{\Gamma }=\lambda ^{(k)}\end{aligned}}}$${\displaystyle {\begin{aligned}-&\Delta u_{i}^{(k)}=f_{i}~~{\text{in}}~~\Omega _{i},\\[1.3ex]&\left.u_{i}^{(k)}\right|_{\partial \Omega }=0,\quad \left.u_{i}^{(k)}\right|_{\Gamma }=\lambda ^{(k)}\end{aligned}}}

for some function λ^(k) on Γ, where λ⁽⁰⁾ is any inexpensive initial guess. We then solve the two Neumann problems

$${\displaystyle {\begin{aligned}-&\Delta \psi _{i}^{(k)}=0~~{\text{in}}~\Omega _{i},\\[1.3ex]&\left.\psi _{i}^{(k)}\right|_{\partial \Omega }=0,\quad \left.\partial _{n_{i}}\psi _{i}^{(k)}\right|_{\Gamma }=\omega \left(\partial _{n_{1}}u_{1}^{(k)}+\partial _{n_{2}}u_{2}^{(k)}\right).\end{aligned}}}$${\displaystyle {\begin{aligned}-&\Delta \psi _{i}^{(k)}=0~~{\text{in}}~\Omega _{i},\\[1.3ex]&\left.\psi _{i}^{(k)}\right|_{\partial \Omega }=0,\quad \left.\partial _{n_{i}}\psi _{i}^{(k)}\right|_{\Gamma }=\omega \left(\partial _{n_{1}}u_{1}^{(k)}+\partial _{n_{2}}u_{2}^{(k)}\right).\end{aligned}}}

We then obtain the next iterate by setting

$${\displaystyle \lambda ^{(k+1)}=\lambda ^{(k)}-\omega \left(\theta _{1}\psi _{1}^{(k)}+\theta _{2}\psi _{2}^{(k)}\right)~~{\text{on}}~\Gamma }$${\displaystyle \lambda ^{(k+1)}=\lambda ^{(k)}-\omega \left(\theta _{1}\psi _{1}^{(k)}+\theta _{2}\psi _{2}^{(k)}\right)~~{\text{on}}~\Gamma }

for some parameters ω, θ₁ and θ₂.

This procedure can be viewed as a Richardson iteration for the iterative solution of the equations arising from the Schur complement method.^[2]

This continuous iteration can be discretized by the finite element method and then solved—in parallel—on a computer. The extension to more subdomains is straightforward, but using this method as stated as a preconditioner for the Schur complement system is not scalable with the number of subdomains; hence the need for a global coarse solve.

• Neumann–Dirichlet method

• Domain decomposition methods