-npc_snes_ | - options prefix of the nonlinear subdomain solver (must be of type NASM) | |
-npc_sub_snes_ | - options prefix of the subdomain nonlinear solves | |
-npc_sub_ksp_ | - options prefix of the subdomain Krylov solver | |
-npc_sub_pc_ | - options prefix of the subdomain preconditioner |
\sum_{i=0}^{N_b}J_b({X^b_{converged}})^{-1}J(X + \sum_{i=0}^{N_b}(X^b_{converged} - X^b))
which is the ASPIN preconditioned matrix. Similar solvers may be constructed by having matrix-free differencing of nonlinear solves per linear iteration, but this is far more efficient when subdomain sparse-direct preconditioner factorizations are reused on each application of J_b^{-1}.
The Krylov method used in this nonlinear solver is run with NO preconditioner, because the preconditioning is done at the nonlinear level, but the Jacobian for the original function must be provided (or calculated via coloring and finite differences automatically) in the Pmat location of SNESSetJacobian() because the action of the original Jacobian is needed by the shell matrix used to apply the Jacobian of the nonlinear preconditioned problem (see above). Note that since the Pmat is not used to construct a preconditioner it could be provided in a matrix-free form. The code for this implementation is a bit confusing because the Amat of SNESSetJacobian() applies the Jacobian of the nonlinearly preconditioned function Jacobian while the Pmat provides the Jacobian of the original user provided function. Note that the original SNES and nonlinear preconditioner preconditioner (see SNESGetNPC()), in this case NASM, share the same Jacobian matrices. SNESNASM computes the need Jacobian in SNESNASMComputeFinalJacobian_Private()
1. | - X. C. Cai and D. E. Keyes, "Nonlinearly preconditioned inexact Newton algorithms", SIAM J. Sci. Comput., 24, 2002. | |
2. | - Peter R. Brune, Matthew G. Knepley, Barry F. Smith, and Xuemin Tu, "Composing Scalable Nonlinear Algebraic Solvers", SIAM Review, 57(4), 2015 |