Combining Aggregation and Classical Algebraic Multigrid in the CPR-AMG Linear Solver

Gratton, S. (University of Toulouse and CERFACS) | Jiranek, P. (CERFACS) | Vasseur, X. (CERFACS) | Henon, P. (Total SA)

OnePetro 

Algebraic multigrid (AMG) represents a class of efficient preconditioners for large and sparse linear systems arising in particular from discretized elliptic partial differential equations. In the context of reservoir simulation, a standard preconditioner for the linear solver is the so-called CPR-AMG in which AMG is applied to an approximated pressure subsystem. In this method, AMG is the bottleneck for scalability because it involves a lot of communications across processors in particular for the setup phase, which constructs the coarse levels according to the coefficients of the Jacobian matrix. The goal of this work is to decrease the overall cost of the CPR-AMG in parallel by combining two different coarsening strategies: the first levels are computed using an aggregation scheme whereas the coarsest levels are treated using a classical point-wise Ruge-Stüben (RS) scheme. AMG preconditioners constructed using classical coarsening schemes are able to achieve good convergence rates of the preconditioned iterative method. The complexities of the multigrid hierarchies can be quite high and thus the classical AMG may be expensive in terms of memory requirements and computational times. Aggregation AMG methods, on the other hand, provide better means of complexity control and consequently the setup time required to construct the preconditioner can be considerably lower. The efficiency of the aggregation AMG methods, however, deteriorates with the increasing problem size although the two-level convergence rates can be very good. To take advantage of both the aggregation and classical AMG, we consider combining both approaches within one hierarchy in order to decrease the setup time of the AMG preconditioner while retaining the convergence properties of the two-level aggregation method.