PHYSICS-BASED BALANCING DOMAIN DECOMPOSITION BY CONSTRAINTS FOR HETEROGENEOUS PROBLEMS

In this work, we present a novel balancing domain decomposition by constraints preconditioner that is robust for multi-material and heterogeneous problems. We start with a well-balanced subdomain partition and, based on an aggregation of elements according to their physical coefficients, we end up with a finer physics-based (PB) subdomain partition. Next, we define geometrical objects (corners, edges, and faces) for this PB partition, and select some of them to enforce subdomain continuity (primal objects). When the physical coefficient in each PB subdomain is constant and the primal objects satisfy a mild condition on the existence of acceptable paths, we can show both theoretically and numerically that the condition number does not depend on the contrast of the coefficient. The constant coefficient condition is computationally feasible for multi-material problems. However, for highly heterogeneous problems, such restriction might result into a large coarse problem. In this case, we propose a relaxed version of the method where we only require that the maximal contrast of the physical coefficient in each PB subdomain is smaller than a predefined threshold. The threshold can be chosen so that the condition number is reasonably small while the size of the coarse problem is not too large. An extensive set of numerical experiments is provided to support our findings. In particular, we show a robustness and a weak scalability analysis up to 8000 cores of the new preconditioner when applied to a 3D heterogeneous problem with more than 260 million degrees of freedom. For the scalability analysis, we have exploited a highly scalable advanced inter-level overlapped implementation of the preconditioner that deals very efficiently with the coarse problem computation.