Adaptive-Multilevel BDDC and its parallel implementation

Abstract

We combine the adaptive and multilevel approaches to the BDDC and formulate a method which allows an adaptive selection of constraints on each decomposition level. We also present a strategy for the solution of local eigenvalue problems in the adaptive algorithm using the LOBPCG method with a preconditioner based on standard components of the BDDC. The effectiveness of the method is illustrated on several engineering problems. It appears that the Adaptive-Multilevel BDDC algorithm is able to effectively detect troublesome parts on each decomposition level and improve convergence of the method. The developed open-source parallel implementation shows a good scalability as well as applicability to very large problems and core counts.Comment: 37 pages, 10 figures, 13 table

Multilevel Balancing Domain Decomposition by Constraints Deluxe Algorithms with Adaptive Coarse Spaces for Flow in Porous Media

“…For domain decomposition methods with continuous pressures, see [29,58,59] and the references therein. We note that our two-level results are equally valid for the FE tearing and interconnecting dual-primal (FETI-DP) method [17], due to the well-known duality between BDDC and FETI-DP [34].After the pioneering work [35], recent research on BDDC methods has focused on controlling the condition number of the preconditioned operators through the adaptive generation of coarse spaces [6,8,23,24,25,37,44,48,64]; these techniques lead to robust preconditioning techniques with tunable rates of convergence; see, in particular, [45]. The adaptive enrichment of the primal space is accomplished by means of solving…”

“…Most previous work on domain decomposition methods for saddle point problems is based on the benign subspace approach [15,38,39,40,42], where the original saddle point is reduced to a positive definite problem in the case of discontinuous pressure discretization spaces. Multilevel extensions of these ideas have been presented in [48,47,57]. For domain decomposition methods with continuous pressures, see [29,58,59] and the references therein.…”

PCBDDC: A Class of Robust Dual-Primal Methods in PETSc

“…Although the proper choice of the scaling operator ensures robustness of the BDDC methods with respect to jumps in the PDE coefficients aligned with the interface, the convergence rate of the associated preconditioned Krylov methods usually deteriorates when such jumps are not aligned with the interface. After the pioneering work [63], in recent years different approaches have been proposed to accommodate arbitrary jumps in the coefficients of elliptic PDEs within BDDC methods [15,19,41,42,44,45,46,69,75].…”

“…The approach proposed in [75] selects face constraints by iteratively solving sparse eigenproblems defined on each pair of subdomains sharing a face and its boundary. Edge primal constraints, which could be obtained as a by-product of the eigensolver, were not considered in the numerical experiments due to a loss of sparsity of the projected operators involved.…”

“…[15,19,41,42,44,45,75,69]. All these techniques increase the arithmetic intensity of the algorithm by properly combining generalized eigenvalue calculations and nearest neighbor communications; the outcome is a robust BDDC preconditioner which is able to reduce the number of iterations of the Krylov solver in a black-box fashion.…”