C. R. Dohrmann. "An approximate BDDC preconditioner", Numerical Linear Algebra with Applications Volume 14, Issue 2, pages 149-168, March 2007  A. Klawonn and O. B. Widlund. "Dual-Primal FETI Methods for Linear Elasticity", https://cs.nyu.edu/dynamic/reports/?year=all  J. Mandel, B. Sousedik, C. R. Dohrmann. "Multispace and Multilevel BDDC", https://arxiv.org/abs/0712.3977  C. Pechstein and C. R. Dohrmann. "Modern domain decomposition methods BDDC, deluxe scaling, and an algebraic approach", Seminar talk, Linz, December 2013, http://people.ricam.oeaw.ac.at/c.pechstein/pechstein-bddc2013.pdf
The matrix to be preconditioned (Pmat) must be of type MATIS.
Currently works with MATIS matrices with local matrices of type MATSEQAIJ, MATSEQBAIJ or MATSEQSBAIJ, either with real or complex numbers.
It also works with unsymmetric and indefinite problems.
Unlike 'conventional' interface preconditioners, PCBDDC iterates over all degrees of freedom, not just those on the interface. This allows the use of approximate solvers on the subdomains.
Approximate local solvers are automatically adapted (see ) if the user has attached a nullspace object to the subdomain matrices, and informed BDDC of using approximate solvers (via the command line).
Boundary nodes are split in vertices, edges and faces classes using information from the local to global mapping of dofs and the local connectivity graph of nodes. The latter can be customized by using PCBDDCSetLocalAdjacencyGraph() Additional information on dofs can be provided by using PCBDDCSetDofsSplitting(), PCBDDCSetDirichletBoundaries(), PCBDDCSetNeumannBoundaries(), and PCBDDCSetPrimalVerticesIS() and their local counterparts.
Constraints can be customized by attaching a MatNullSpace object to the MATIS matrix via MatSetNearNullSpace(). Non-singular modes are retained via SVD.
Change of basis is performed similarly to  when requested. When more than one constraint is present on a single connected component (i.e. an edge or a face), a robust method based on local QR factorizations is used. User defined change of basis can be passed to PCBDDC by using PCBDDCSetChangeOfBasisMat()
The PETSc implementation also supports multilevel BDDC . Coarse grids are partitioned using a MatPartitioning object.
Adaptive selection of primal constraints  is supported for SPD systems with high-contrast in the coefficients if MUMPS or MKL_PARDISO are present. Future versions of the code will also consider using PASTIX.
An experimental interface to the FETI-DP method is available. FETI-DP operators could be created using PCBDDCCreateFETIDPOperators(). A stand-alone class for the FETI-DP method will be provided in the next releases. Deluxe scaling is not supported yet for FETI-DP.
Options for Dirichlet, Neumann or coarse solver can be set with
-pc_bddc_dirichlet_ -pc_bddc_neumann_ -pc_bddc_coarse_e.g -pc_bddc_dirichlet_ksp_type richardson -pc_bddc_dirichlet_pc_type gamg. PCBDDC uses by default KPSPREONLY and PCLU.
When using a multilevel approach, solvers' options at the N-th level (N > 1) can be specified as
-pc_bddc_dirichlet_lN_ -pc_bddc_neumann_lN_ -pc_bddc_coarse_lN_Note that level number ranges from the finest (0) to the coarsest (N). In order to specify options for the BDDC operators at the coarser levels (and not for the solvers), prepend -pc_bddc_coarse_ or -pc_bddc_coarse_l to the option, e.g.
-pc_bddc_coarse_pc_bddc_adaptive_threshold 5 -pc_bddc_coarse_l1_pc_bddc_redistribute 3will use a threshold of 5 for constraints' selection at the first coarse level and will redistribute the coarse problem of the first coarse level on 3 processors
Contributed by Stefano Zampini