Algebraic preconditioning analysis of the multilevel block incremental unknowns method for anisotropic elliptic operators
Condition number of the block incremental unknowns (BIU) matrix associated to anisotropic operator epsilon partial derivative(2)/partial derivative x(2) + partial derivative(2)/partial derivative y(2) with 0 < epsilon << 1 is analyzed; more general second-order anisotropic elliptic operators are also considered. Theoretical analyses show that the condition number of the BIU matrix is bounded by c . (h(-1) + epsilon h(-2)) instead of O(h(-2)) with usual nodal unknowns where h is the mesh size. In addition, we introduce a diagonal preconditioner such that the condition number of the BIU matrix is further reduced to O(1 + 4 epsilon pi(-2)h(-2)), which means that the condition number is optimal if epsilon = O(h(2)). Numerical experiments are performed to different examples and parameters, numerical results verify our theoretical analysis. (C) 2012 Elsevier Ltd. All rights reserved.
preconditioning block incremental unknowns condition number diagonal preconditioner anisotropic problem