Reduced order modeling for elliptic problems with high contrast diffusion coefficients

This paper addresses reduced order modeling for parametric elliptic PDEs with high contrast diffusion coefficients. Unlike common assumptions, the problem isn't uniformly elliptic due to arbitrarily high contrast. The authors construct reduced model spaces providing uniform approximation of all solutions with contrast-independent relative error estimates. These estimates, while sub-exponential in reduced model dimension, exhibit the curse of dimensionality as the number of subdomains increases. Similar estimates are derived for Galerkin projection and inverse problems (state and parameter estimation). A key aspect is analyzing the convergence to limit solutions in stiff problems as diffusion tends to infinity in certain domains.

Albert Cohen, Wolfgang Dahmen, Matthieu Dolbeault, Agustin Somacal