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

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

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

$u_1 = u_2, \qquad \partial_nu_1 = \partial_nu_2$

where n is the unit normal vector to Γ.

An iterative method for approximating each u_i satisfying the matching conditions is to first solve the decoupled problems (i=1,2)

$-\Delta u_i^{(k)} = f_i, \qquad u_i^{(k)}|_{\partial\Omega} = 0, \quad u^{(k)}_i|_\Gamma = \lambda^{(k)}$

for some function λ^(k) on Γ. We then solve the two Neumann problems

$-\Delta\psi_i^{(k)} = 0, \qquad \psi_i^{(k)}|_{\partial\Omega} = 0, \quad \partial_n\psi_i^{(k)} = \partial_nu_1^{(k)} - \partial_nu_2^{(k)}.$

We then obtain the next iterate by setting

$\lambda^{(k%2B1)} = \lambda^{(k)} - \omega(\theta_1\psi_1^{(k)}|_\Gamma - \theta_2\psi_2^{(k)}|_\Gamma)$

for some parameters ω, θ₁ and θ₂.

This procedure can be viewed as a Richardson extrapolation 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.

References

^ A. Klawonn and O. B. Widlund, FETI and Neumann–Neumann iterative substructuring methods: connections and new results, Comm. Pure Appl. Math., 54 (2001), pp. 57–90.
^ A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford Science Publications 1999.

Numerical partial differential equations

Finite difference methods	Heat Equation and related: FTCS scheme · Crank–Nicolson method Hyperbolic: Lax–Friedrichs method · Lax–Wendroff method · MacCormack method · Upwind scheme · Other: Alternating direction implicit method · Finite-difference time-domain method

Finite volume methods	Godunov's scheme · High-resolution scheme · MUSCL scheme · AUSM · Riemann solver

Finite element methods	hp-FEM · Extended finite element method · Discontinuous Galerkin method · Spectral element method · Meshfree methods · Mortar methods

Other methods	Spectral method · Pseudospectral method · Method of lines · Multigrid methods · Collocation method · Level set method · Boundary element method · Immersed boundary method · Analytic element method · Particle-in-cell · Isogeometric analysis

Domain decomposition methods	Schur complement method · Fictitious domain method · Schwarz alternating method · Additive Schwarz method · Abstract additive Schwarz method · Neumann–Dirichlet method · Neumann–Neumann methods · Poincaré–Steklov operator · Balancing domain decomposition · BDDC · FETI · FETI-DP

Neumann–Neumann methods

See also

References