Linear and nonlinear substructured Restricted Additive Schwarz iterations and preconditioning

Abstract

Iterative substructuring Domain Decomposition (DD) methods have been extensively studied, and they are usually associated with nonoverlapping decompositions. It is less known that classical overlapping DD methods can also be formulated in substructured form, i.e., as iterative methods acting on variables defined exclusively on the interfaces of the overlapping domain decomposition. We call such formulations substructured domain decomposition methods. We introduce here a substructured version of Restricted Additive Schwarz (RAS) which we call SRAS. We show that RAS and SRAS are equivalent when used as iterative solvers, as they produce the same iterates, while they are substantially different when used as preconditioners for GMRES. We link the volume and substructured Krylov spaces and show that the iterates are different by deriving the least squares problems solved at each GMRES iteration. When used as iterative solvers, SRAS presents computational advantages over RAS, as it avoids computations with matrices and vectors at the volume level. When used as preconditioners, SRAS has the further advantage of allowing GMRES to store smaller vectors and perform orthogonalization in a lower dimensional space. We then consider nonlinear problems, and we introduce SRASPEN (Substructured Restricted Additive Schwarz Preconditioned Exact Newton), where SRAS is used as a preconditioner for Newton’s method. In contrast to the linear case, we prove that Newton’s method applied to the preconditioned volume and substructured formulation produces the same iterates in the nonlinear case. Next, we introduce two-level versions of nonlinear SRAS and SRASPEN. Finally, we validate our theoretical results with numerical experiments.

A Nonlinear Elimination Preconditioned Inexact Newton Algorithm

“…Left nonlinear preconditioners lead to a system with the same root as the original system, which is solved by an outer Jacobian-free Newton method [26]. Examples include the ad-ditive (and multiplicative) Schwarz preconditioned inexact Newton methods ASPIN (MSPIN) [1,3,5,22,38] and two-level ASPIN [6,34] and MSPIN [28,29,30], and the restricted nonlinear Schwarz preconditioners RASPEN [12,16] and SRASPEN [9]. As with linear preconditioning, a left-preconditioned Jacobian is generally not formed explicitly, it being typically much denser than the original; only the matrix-vector multiplication is provided for Krylov subspace methods.…”

Asynchronous domain decomposition methods for nonlinear PDEs

“…We note that earlier work has been done on using RAS as a preconditioner; cf. [12,17]. We refer the reader also to the recent work [28] for two-level RAS preconditioners using additive/multiplicative coupling of coarse spaces, to [13] for substructured coarse correction, and to [24] for asynchronous multiplicative coarse correction.…”