Domain Decomposition Methods — Domain decomposition divides a computational domain into subdomains assigned to different processors, enabling parallel solution of partial differential equations and other spatially-structured problems by combining local solutions with boundary exchange communication.

Spatial Partitioning Strategies — Dividing the domain determines communication and load balance:

• Regular Grid Decomposition — structured grids are divided into rectangular blocks along coordinate axes, producing simple communication patterns with predictable load distribution
• Recursive Bisection — the domain is recursively split along the longest dimension, creating balanced partitions that adapt to irregular domain shapes and non-uniform computational density
• Graph-Based Partitioning — tools like METIS and ParMETIS model the mesh as a graph and partition it to minimize edge cuts while maintaining balanced vertex weights across partitions
• Space-Filling Curves — Hilbert or Morton curves map multi-dimensional domains to one-dimensional orderings that preserve spatial locality, enabling simple partitioning with good communication characteristics

Ghost Cell Communication — Boundary data exchange enables local computation:

• Halo Regions — each subdomain is extended with ghost cells that mirror boundary values from neighboring subdomains, providing the data needed for stencil computations near partition boundaries
• Exchange Protocols — at each time step or iteration, processors exchange updated ghost cell values with their neighbors using point-to-point MPI messages or one-sided communication
• Halo Width — the number of ghost cell layers depends on the stencil width, with wider stencils requiring deeper halos and proportionally more communication per exchange
• Asynchronous Exchange — overlapping ghost cell communication with interior computation hides latency by initiating non-blocking sends and receives before computing interior points

Non-Overlapping Domain Decomposition — Subdomains share only boundary interfaces:

• Schur Complement Method — eliminates interior unknowns to form a reduced system on the interface, which is solved iteratively before recovering interior solutions independently
• Balancing Domain Decomposition — a preconditioner that ensures the condition number of the interface problem grows only polylogarithmically with the number of subdomains
• FETI Method — the Finite Element Tearing and Interconnecting method uses Lagrange multipliers to enforce continuity at subdomain interfaces, naturally producing a parallelizable dual problem
• Iterative Substructuring — alternates between solving local subdomain problems and updating interface conditions until the global solution converges

Overlapping Domain Decomposition — Subdomains share overlapping regions for improved convergence:

• Additive Schwarz Method — all subdomain problems are solved simultaneously and their solutions are combined, providing natural parallelism with convergence rate depending on overlap width
• Multiplicative Schwarz Method — subdomain problems are solved sequentially using the latest available boundary data, converging faster but offering less parallelism than the additive variant
• Restricted Additive Schwarz — each processor only updates its owned portion of the overlap region, reducing communication while maintaining convergence properties
• Coarse Grid Correction — adding a coarse global problem that captures long-range interactions dramatically improves convergence, preventing the iteration count from growing with the number of subdomains

Domain decomposition methods are the primary approach for parallelizing PDE solvers in computational science, with their mathematical framework providing both practical scalability and theoretical convergence guarantees for large-scale simulations.