On POD-based Deflation Vectors for DPCG applied to porous media problems

An important question when using deflation techniques is how to find good deflation vectors, which lead to a decrease in the number of iterations and a small increase in the required computing time per iteration. In this paper, we propose the use of deflation vectors based on a POD-reduced set of snapshots. We investigate convergence and the properties of the resulting methods. Finally, we illustrate these theoretical results with numerical experiments. We consider compressible and incompressible single-phase flow in a layered model with variations in the permeability layers up to 103 and the SPE 10 benchmark model with a contrast in permeability coefficients of 107. Using deflation for the incompressible problem, we reduce the number of iterations to 1 or 2 iterations. With deflation, for the compressible problem, we reduce up to ∼80% the number of iterations when compared with the only-preconditioned solver.