Domain Decomposition and Schwarz Methods

Solvers that split a large problem into overlapping or non-overlapping subdomains, solving each locally and coordinating across interfaces.

Divide and conquer for PDEs

Domain decomposition splits the computational domain into subdomains, solves a smaller problem on each, and iterates to enforce consistency where subdomains meet. The approach maps naturally onto parallel hardware: each processor owns a subdomain, and communication is limited to interface data. This is central to running large plasma and structural simulations on thousands of cores.

The classical Schwarz alternating method

Hermann Schwarz's original method used two overlapping regions: solve on the first with a guessed boundary value from the second, then solve on the second using the updated first solution, and alternate. The overlap carries information between subdomains. Larger overlap gives faster convergence but more redundant work.

Additive and multiplicative variants

• Multiplicative Schwarz solves subdomains sequentially, each using the latest neighbor data (like Gauss-Seidel across subdomains)
• Additive Schwarz solves all subdomains simultaneously from the same iterate (like Jacobi), making it fully parallel
• Restricted additive Schwarz avoids double-counting in overlap regions and often converges faster

The coarse space problem

One-level Schwarz methods have a flaw: convergence degrades as the number of subdomains grows, because information travels only one subdomain per iteration. Adding a coarse space (a global, low-resolution problem) fixes this by propagating information across the whole domain each step, restoring scalability. This two-level structure makes domain decomposition closely related to multigrid.

In practice

Schwarz methods are most often used as parallel preconditioners for Krylov solvers rather than standalone iterations. The FETI and balancing methods are non-overlapping cousins that enforce continuity through Lagrange multipliers, widely used in large finite-element structural and electromagnetic analyses.