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Problems with localized nonhomogeneous material properties present well-knownchallenges for numerical simulations. In particular, such problems may featurelarge differences in length scales, causing difficulties with meshing andpreconditioning. These difficulties are increased if the region of localizeddynamics changes in time. Overlapping domain decomposition methods, which splitthe problem at the continuous level, show promise due to their ease ofimplementation and computational efficiency. Accordingly, the present work aimsto further develop the mathematical theory of such methods at both thecontinuous and discrete levels. For the continuous formulation of the problem,we provide a full convergence analysis. For the discrete problem, we show howthe described method may be interpreted as a Gauss-Seidel scheme or as aNeumann series approximation, establishing a convergence criterion in terms ofthe spectral radius of the system. We then provide a spectral scaling argumentand provide numerical evidence for its justification.

A theoretical and numerical analysis of a Dirichlet-Neumann domain decomposition method for diffusion problems in heterogeneous media