Schwarz alternating method

Remarks

Let $P_i$ be the $a$-orthogonal projection onto $V_i$. The errors satisfy $u^{n+1} - u = (I - P_2)(I - P_1)(u^n - u)$, a product of the projections onto $V_1^\perp$ and $V_2^\perp$. Since $V_1 + V_2 = V$ is closed, $V_1^\perp \cap V_2^\perp = \set{0}$ and the angle between $V_1^\perp$ and $V_2^\perp$ is positive, which gives $\rho = \norm{(I - P_2)(I - P_1)}_a < 1$. If $V_1 + V_2$ is only dense in $V$, von Neumann's theorem on alternating projections still gives $u^n \to u$, but possibly without a geometric rate.

The original setting is the Dirichlet problem $-\Delta u = f$ in $\Omega$, $u = 0$ on $\partial \Omega$, where $\Omega = \Omega_1 \cup \Omega_2$ is a union of two overlapping domains. Take $V = H^1_0(\Omega)$ (a Sobolev space), $a(u, v) = \int_\Omega \nabla u \cdot \nabla v \, dx$, and $V_i = H^1_0(\Omega_i)$ extended by zero. Then each half step solves the Dirichlet problem on $\Omega_i$ with boundary values taken from the current iterate on $\partial \Omega_i \cap \Omega$. A partition of unity subordinate to the overlap shows $V_1 + V_2 = V$ when the overlap has positive width. The rate $\rho$ tends to $1$ as the overlap shrinks.

Going deeper

For a student

The method is the prototype of domain decomposition methods: a large problem is split into problems on subdomains that are cheaper to solve. In the abstract form it is a block Gauss–Seidel iteration on the splitting $V = V_1 + V_2$, and the error operator is a product of orthogonal projections. The additive Schwarz method is the Jacobi-type variant, which solves on all pieces at once and is used as a preconditioner. The convergence rate depends on the overlap, which is why practical methods add a coarse space.