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Transition Of Algebraic Multiscale To Algebraic Multigrid

Summary Algebraic Multiscale (AMS) is a recent development for the construction of efficient linear solvers in certain reservoir simulations. It employs analytical upscaling ideas to coarsen the respective linear system and provides a high amount of inherent parallelism. However, it has the drawback that it can currently only be applied to problems for a single scalar physical unknown, e.g., a pressure sub-problem. Moreover, typical AMS exploits a structured grid and results in a two-level scheme only. Generalizing the AMS approach to overcome these limitations requires substantial efforts and is not straightforward. To exploit the benefits of AMS, however, we integrate its core ideas in an Algebraic Multigrid (AMG) method. Thus, all results and techniques from the well-established AMG are directly available in conjunction with (core ingredients of) AMS. This holds regarding multilevel usage and applicability for unstructured problems. But it also holds for the System-AMG approach that allows to consider additional thermal and mechanical unknowns. In this paper, we identify the algorithmic similarities between both approaches, AMS and AMG. In fact, the basic coarsening idea of AMS corresponds to the so-called aggregative AMG approach. However, plain aggregative AMG suffers from the drawback of simplified transfer operation, or interpolation, within the hierarchy. By the integration of the AMS transfer we overcome this limitation and significantly improve the robustness of the aggregative AMG; especially in cases with inhomogeneous material coefficients. Yet, the method is not as robust as classical AMG, though. However, the setup phase is significantly simplified. Moreover, we adjust the AMS-like interpolation to work purely algebraically, independent of any grid structure. This involves certain compromises to the original AMS idea. However, it can now be used as an interpolation in any aggregative AMG approach. An additional advantage of our approach is the improved controllability of the hierarchy’s operator complexity, i.e., its memory consumption. This is especially important with increasing density of the matrix stencils, e.g., in (geo)mechanics or data science.