A numerical study of a semi-algebraic multilevel preconditioner for the local discontinuous Galerkin method

In this work, we consider the local discontinuous Galerkin (LDG) method applied to second-order elliptic problems arising in the modeling of single-phase flows in porous media. It has been recently proven that the spectral condition number of the stiffness matrix exhibits an asymptotic behavior of O(h(-2)) on structured and unstructured meshes, where It is the mesh size. Thus, efficient preconditioners are mandatory. We present a semi-algebraic multilevel preconditioner for the LDG method using local Lagrange-type interpolatory basis functions. We show, numerically, that its performance does not degrade, or at least the number of iterations increases very slowly, as the number of unknowns augments. The preconditioner is tested on problems with high jumps in the coefficients, which is the typical scenario of problems arising in porous media. Copyright (C) 2007 John Wiley & Sons, Ltd.