  Our research focuses on the development of domain decomposition
preconditioners tailored for second-order elliptic partial differential
equations. Our approach addresses two major challenges simultaneously: i)
effectively handling coefficients with high-contrast and multiscale properties,
and ii) accommodating irregular domains in the original problem, the coarse
mesh, and the subdomain partition. The robustness of our preconditioners is
crucial for real-world applications, such as the efficient and accurate
modeling of subsurface flow in porous media and other important domains. The
core of our method lies in the construction of a suitable partition of unity
functions and coarse spaces utilizing local spectral information. Leveraging
these components, we implement a two-level additive Schwarz preconditioner. We
demonstrate that the condition number of the preconditioned systems is bounded
with a bound that is independent of the contrast. Our claims are further
substantiated through selected numerical experiments, which confirm the
robustness of our preconditioners.
