On overlapping domain decomposition methods for high-contrast multiscale problems
For researchers working on numerical methods for multiscale problems, this work offers a preconditioner with near-optimal condition number, though it is an incremental improvement over existing spectral coarse space methods.
This paper reviews overlapping domain decomposition methods for high-contrast multiscale problems and proposes a new method that achieves a condition number close to 1, independent of contrast and small scales, by constructing minimal-dimensional coarse spaces using spectral information.
We review some important ideas in the design and analysis of robust overlapping domain decomposition algorithms for high-contrast multiscale problems and propose a domain decomposition method better performance in terms of the number of iterations. The main novelty of our approaches is the construction of coarse spaces, which are computed using spectral information of local bilinear forms. We present several approaches to incorporate the spectral information into the coarse problem in order to obtain minimal coarse space dimension. We show that using these coarse spaces, we can obtain a domain decomposition preconditioner with the condition number independent of contrast and small scales. To minimize further the number of iterations until convergence, we use this minimal dimensional coarse spaces in a construction combining them with large overlap local problems that take advantage of the possibility of localizing global fields orthogonal to the coarse space. We obtain a condition number close to 1 for the new method. We discuss possible drawbacks and further extensions.