A BDDC Algorithm with Enriched Coarse Spaces for Two-Dimensional Elliptic Problems with Oscillatory and High Contrast Coefficients

Abstract

A balancing domain decomposition by constraints (BDDC) algorithm with enriched coarse spaces is developed and analyzed for two-dimensional elliptic problems with oscillatory and high contrast coefficients. To obtain a robust algorithm based on the classical BDDC framework for conforming finite element methods, a set of enriched primal unknowns is constructed. The enriched component of the primal unknowns is chosen to reflect the local structures of the coefficient by solving two types of generalized eigenvalue problems. These problems are solved locally for every two adjacent subdomains, so that the computations of the enriched primal unknowns are very efficient. Given a tolerance, dominant eigenfunctions with eigenvalues larger than this tolerance are precomputed and used to form coarse basis functions. The resulting condition number by using this enriched coarse space is proved to be bounded above by this tolerance and the maximum number of edges per subdomain, independent of the contrast of the given coefficient. Furthermore, numerical results are presented for various model problems and various choices of primal unknowns to show the performance of the proposed algorithm.