An iterative solving framework for local fine space of balancing domain decomposition

In this paper, we propose an improved balancing domain decomposition (BDD) method, developed within an iterative solving framework, for large-scale symmetric positive-definite sparse systems. Traditional BDD-NN approaches, which combine Neumann–Neumann (NN) preconditioners with direct solvers for local fine space problems, encounter computational limitations due to the singularity of their coefficient matrix. To address this limitation, we introduce a regularization technique and replace direct solvers with iterative alternatives. Two categories of iterative methods—matrix splitting techniques, such as Gauss Seidel (GS) and Successive Over Relaxation (SOR), and Krylov subspace methods, including the Full Orthogonalization Method (FOM), Generalized Minimum Residual (GMRES), and Minimum Residual Method (MINRES)—are incorporated into the BDD framework to solve local fine space problems effectively. We systematically formulate the resulting BDD-GS, BDD-SOR, BDD-FOM, BDD-GMRES, and BDD-MINRES methods and theoretically establish the convergence of the corrected local problems for these proposed iterative strategies. We evaluate performance using 10 nodes, each with 64 processors, focusing on linear systems derived from the Poisson equation. Experimental results show that the proposed BDD-type iterative methods achieve enhanced parallel scalability, maintain numerical scalability, and reduce computational time compared to the conventional BDD-NN for large-scale Schur complement systems. These results confirm the effectiveness of iterative BDD variants in addressing large-scale problems.