Algebraic Multiscale Linear Solver for Heterogeneous Elliptic Problems a Report Submitted to the Department of Energy Resources Engineering of Stanford University in Partial Fulfillment of the Requirements for the Degree of Master of Science

An Algebraic Multiscale Solver (AMS) for the pressure system of equations arising from incompressible flow in heterogeneous porous media is developed. The algorithm allows for several independent preconditioning stages to deal with the full spectrum of errors. In addition to the fine-scale system of equations, AMS requires information about the superimposed (dual) coarse grid, which is used for the construction of a wirebasket ordering. The primal coarse grid is used in the construction of a conservative coarse-scale operator and in the reconstruction of a conservative fine-scale velocity field as the last step of the solution process. The convergence properties of AMS are studied for various combinations including (1) the MultiScale FiniteElement (MSFE) method, (2) the MultiScale Finite-Volume (MSFV) approach, (3) Correction Functions (CF), (4) Block ILU with zero fill-in (BILU), and (5) point-wise ILU with zero fill-in (ILU). The reduced-problem boundary condition, which is used for localization, is investigated and improvements are proposed. For a wide range of test cases, including the highly heterogeneous (with over a million cells) SPE 10 permeability field, the performance of the different preconditioning options is analyzed. It is found that the best overall performance is obtained by combining MSFE and ILU as the global and local preconditioners, respectively. Comparison between AMS and the widely used SAMG solver illustrates that AMS and SAMG are comparable, especially for very large heterogeneous problems.