Balancing domain decomposition method
Balancing domain decomposition method (BDD) is an iterative technique for solving large symmetric positive definite linear systems that come from the finite element method. Each iteration combines solutions from small, non-overlapping subdomains with a simpler coarse problem built from the subdomains’ nullspaces. The method only needs the ability to solve subdomain problems, not to access their full matrices, so it works well when you only have the solution operators, such as in oil reservoir simulations.
Originally, BDD works best for second‑order problems like common elasticity in 2D and 3D. For fourth‑order problems (for example, plate bending), the coarse problem must be enhanced with special basis functions to enforce continuity at subdomain corners, which increases the cost.
The BDDC method is a related approach that uses the same corner basis functions but in an additive way rather than a multiplicative one. The dual counterpart to BDD is FETI, which enforces subdomain agreement using Lagrange multipliers. The basic BDD and FETI are not mathematically the same, but a robust version of FETI can have essentially the same performance as BDD.
In this framework, the overall system solved by BDD can be viewed as the Schur complement on the subdomain interfaces after interior unknowns are eliminated. Because the BDD preconditioner relies on solving Neumann problems on every subdomain, it belongs to the Neumann–Neumann family of methods (where a Neumann problem is solved on both sides of the subdomain interface).
The coarse space is simple at first—constants on each subdomain averaged across interfaces. More generally, the coarse space can be chosen to include the nullspace of the local problems, i.e., the nontrivial patterns that do not change the interior of a subdomain.
This page was last edited on 3 February 2026, at 01:43 (CET).