An overlapping domain decomposition method based on solution-transfer operators

Abstract

An overlapping domain decomposition method is described for variable-coefficient elliptic boundary value problems on domains that can be decomposed into slabs or shells. The method represents the global solution through its traces on internal interfaces, coupled by local Dirichlet solution transfer operators posed on overlapping double slab domains. The key observation is that these interface maps act between separated interfaces, and as such can be written as smooth-kernel integral operators. The resulting global equilibrium system is Fredholm second kind and, unlike non-overlapping formulations, requires no same-interface Dirichlet-to-Neumann or other interface maps with singular kernels. This makes its off-diagonal blocks highly amenable to hierarchical low-rank compression. The formulation admits complementary continuum and discrete interpretations. At fixed slab width, the method is stable under discretization, sufficiently accurate local solves and compression. At the discrete level, the system can be interpreted as a block Jacobi preconditioned Schur complement system. For compatible SPD discretizations satisfying a standard stable-splitting assumption, the symmetrically scaled interface matrix satisfies an energy-norm condition-number bound that depends on the slab width but is uniform with respect to the local resolution. The formulation is implemented using high-order local solvers and hierarchical compression based on randomized sampling. Numerical experiments report iteration counts, accuracy, and compressibility for 2D and 3D elliptic, nonsymmetric, and oscillatory problems with as many as 28 million degrees of freedom.