Preconditioning analysis of nonuniform incremental unknowns method for two dimensional elliptic problems

For the linear system obtained by discretizing two dimensional elliptic boundary value problems on nonuniform meshes, the condition number of the coefficient matrix preconditioned by nonuniform incremental unknowns (NUIUs) method, abbreviated as NUIUs matrix, is carefully analyzed. Comparing to the original coefficient matrix, the condition number of the NUIUs matrix is reduced from O(ad) to O(d2) with a≥4 and d being the level of discretization. Numerical experiments are performed, respectively, on three types of nonuniform meshes to verify the correctness of our theoretical analysis and test the preconditioning efficiency of the NUIUs method.