Parallel Implementation of Multilevel BDDC

Abstract

In application of the Balancing Domain Decomposition by Constraints (BDDC) to a case with many substructures, solving the coarse problem exactly becomes the bottleneck which spoils scalability of the solver. However, it is straightforward for BDDC to substitute the exact solution of the coarse problem by another step of BDDC method with subdomains playing the role of elements. In this way, the algorithm of three-level BDDC method is obtained. If this approach is applied recursively, multilevel BDDC method is derived. We present a detailed description of a recently developed parallel implementation of this algorithm. The implementation is applied to an engineering problem of linear elasticity and a benchmark problem of Stokes flow in a cavity. Results by the multilevel approach are compared to those by the standard (two-level) BDDC method.

Adaptive-Multilevel BDDC and its parallel implementation

“…It was shown in that reference, that while convergence of BDDC with the standard choice of arithmetic averages on faces quickly deteriorates with increasing contrast, adaptive version of the algorithm is capable of maintaining good convergence also for large values of contrast, at the cost of quite expensive set-up phase. The multilevel approach (without adaptivity), although it may lead to faster solution, suffers from exponentially growing condition number and related number of iterations, as reported in [26], or recently in [32]. Here, we investigate the effect of constraints adaptively generated at higher levels in the multilevel algorithm.…”

“…Finally, we introduce the space W st Γ = W st ∩ W st Γ . Now the generalized eigenvalue problem (32) becomes a localized problem to find w ∈ W st Γ such that…”

“…A parallel instance of MUMPS is also invoked for factorization and repeated solution of the final coarse problem at the top level. More details on implementation of the (non-adaptive) multilevel approach in BDDCML can be found in [32].…”

“…The multilevel approach (without adaptivity), although it may lead to faster solution, suffers from exponentially growing condition number and related number of iterations, as reported in [26], or recently in [32]. Here, we investigate the effect of constraints adaptively generated at higher levels in the multilevel algorithm.…”

Choosing the Subregions in Three-Level FROSch Preconditioners

“…, corresponding to the overlapping subregions Ω ′ i0 , as well as coarse basis functions Φ 0 , spanning the (R)GDSW coarse space V 00 on the third level, are needed to build the three-level preconditioners; see [9,10,8] for more details. Note that the three-level (R)GDSW approach is related to three-level or multilevel BDDC methods [16,14,2,15]. Our undertaking is, software-wise, of course related to other important scalable implementations of Schwarz methods, e.g., [12].…”