BDdc

BDDC stands for Balancing Domain Decomposition by Constraints, a nonoverlapping domain decomposition preconditioner used to accelerate iterative solvers for large sparse linear systems arising from finite element discretizations. It partitions the computational domain into subdomains, solves local problems, and constructs a coarse-level problem to couple subdomains through carefully chosen continuity constraints. The resulting preconditioner improves robustness and scalability, particularly for problems with heterogeneous material properties or complex geometries.

Core idea: impose continuity by enforcing constraints across subdomain interfaces (primal degrees of freedom). The method defines a set of primal constraints (such as average values on subdomain faces or edges and at subdomain corners) and ensures that interface unknowns are balanced. Local solves on each subdomain are performed with these constraints, and a global coarse problem enforces consistency. The preconditioned system accelerates convergence of Krylov solvers.

Relation and history: BDDC was introduced by Dohrmann in 2003 as a member of the family of FETI-DP methods, emphasizing balancing constraints to achieve robust convergence regardless of coefficient contrasts. It is mathematically related to FETI-DP but uses a primal coarse space that can be simpler to implement in some settings. Convergence theory has been established for a wide class of second-order elliptic problems and many FE discretizations.

Implementation and use: widely implemented in parallel finite element software. Suitable for 2D and 3D problems, including linear elasticity, diffusion, and Stokes-like systems. The method scales well with the number of processors, as most work is done independently on subdomains, with a coarse correction step that remains manageable. Variants and extensions include adaptive constraint selection and multilevel BDdc formulations.

Variants: extensions to nearly incompressible materials, heterogeneous coefficients, and multilevel or GenEO-enhanced BDdc aim to improve performance on challenging problems.