A Two-level Overlapping CEM-DDM Preconditioner For General Mesh Partition

An economic two-level overlapping Schwarz method based on constraint energy minimizing(CEM)multiscale coarse basis functions is proposed for the linear finite element discretization of second-order elliptic problems with highly oscillatory and high contrast coefficients on structured and unstructured grids respectively.We also call the mentioned coarse basis functions as the coarse basis functions based on the generalized eigenvalue problem and the mentioned preconditoner as overlapping CEM-DDM preconditioner.Under the structured grid,we firstly classify the nodes in the coarse subregion and develop an optimization algorithm for generating the coarse matrix.Then,we presented some numerical experiments about the CG method based on overlapping CEM-DDM preconditioner(CEM-DDM-CG for short)and the CG method based on standard coarse space(STD-DDM-CG for short).Under the unstructured grid,we firstly describe the data structure of the unstructured grid,and generate the mesh of the overlapping sub-region and the interior sub-mesh of the non-overlapping sub-region.In the process of generating the former mesh,we adopt a special tecnique which is called interface extension in order to avoid the suspension point.Then,a fast algorithm is designed for generating the coarse matrices by classifying the nodes in the coarse sub-region.Finally,the numerical experiments about CEM-DDM-CG are given.The numerical results show that iterations of CEM-DDM-CG is weakly dependent on the jump distribution of the coefficient and the mesh size for both structured and unstructured meshes,and the algorithmic scalability of CEMDDM-CG is better than STD-DDM-CG on structured mesh,especially for the case of random jump coefficients.