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

TL;DR

This paper develops a comprehensive mathematical framework for a Dirichlet-Neumann domain decomposition method applied to diffusion problems in heterogeneous media, including convergence analysis and spectral considerations.

Contribution

It advances the theory of overlapping domain decomposition methods at both continuous and discrete levels, with new convergence criteria and spectral analysis.

Findings

Full convergence analysis for the continuous formulation.

Interpretation of the discrete method as Gauss-Seidel or Neumann series.

Numerical evidence supporting spectral scaling arguments.

Abstract

Problems with localized nonhomogeneous material properties present well-known challenges for numerical simulations. In particular, such problems may feature large differences in length scales, causing difficulties with meshing and preconditioning. These difficulties are increased if the region of localized dynamics changes in time. Overlapping domain decomposition methods, which split the problem at the continuous level, show promise due to their ease of implementation and computational efficiency. Accordingly, the present work aims to further develop the mathematical theory of such methods at both the continuous and discrete levels. For the continuous formulation of the problem, we provide a full convergence analysis. For the discrete problem, we show how the described method may be interpreted as a Gauss-Seidel scheme or as a Neumann series approximation, establishing a convergence criterion in terms of the spectral radius of the system. We then provide a spectral scaling argument and provide numerical evidence for its justification.