From Domain Decomposition to Homogenization Theory
Provides a new perspective on homogenization theory for mathematicians and engineers working on multiscale PDEs, but the result itself is a rediscovery of known theory.
The paper rederives a classical homogenization result for linear elliptic problems with periodic coefficients using domain decomposition and finite elements, avoiding traditional analytical methods. The approach generalizes to non-periodic and multiscale problems.
This paper rediscovers a classical homogenization result for a prototypical linear elliptic boundary value problem with periodically oscillating diffusion coefficient. Unlike classical analytical approaches such as asymptotic analysis, oscillating test functions, or two-scale convergence, the result is purely based on the theory of domain decomposition methods and standard finite elements techniques. The arguments naturally generalize to problems far beyond periodicity and scale separation and we provide a brief overview on such applications.