An Algebraic Multilevel Method with Applications

Abstract

In this chapter, we describe an algebraic multilevel linear-system solver that is defined only in terms of the elements in the coefficient matrix; no underlying PDE or grid is assumed. Still, the applications of the method are mostly to problems arising from the discretization of scalar elliptic PDEs on unstructured grids. Under some algebraic assumptions such as diagonal dominance of the coefficient matrices, we provide an a-posteriori upper bound for the condition number of the V(0,0)-cycle. For diffusion problems with variable and discontinuous coefficients, it is indicated that this upper bound exhibits the moderate growth of O(L ³), where L is the number of levels used. Furthermore, this bound is independent of the discontinuities in the coefficients in the PDE and the meshsize. The method is applied to difficult problems such as highly anisotropic diffusion equation, the Maxwell equations on 3-D staggered grid, and singularly-perturbed convection-diffusion equation.