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Nonlinear Domain Decomposition Methods: Analysis and Industrial Applications

EDMI (Mathématiques et Informatique) – [Subject to defense authorization]

This thesis focuses on the design, analysis, and high performance implementation of parallel nonlinear solvers for large-scale discretizations of nonlinear partial differential equations. These problems yield large nonlinear systems whose cost is dominated by repeated linearizations and Krylov solves. Because linear domain decomposition cannot achieve perfect parallel efficiency at scale, its overhead is paid at every nonlinear iteration, motivating methods that act directly on the nonlinear convergence loop. We develop nonlinear domain decomposition preconditioners for the Newton method, with emphasis on RASPEN and its substructured variant SRASPEN. A theoretical analysis explains how nonlinear subdomain corrections condense the nonlinearity onto interface unknowns and improve the contraction properties of the induced Newton mapping. We then introduce nonlinear two-level preconditioning through an algebraic coarse correction computed by a nonlinear subspace iteration. We focus on a multiplicative approach and identify conditions on the coarse space that enhance favorable spectral properties of the initial two-level Jacobian. Practical constructions follow from local singular value problems posed on the interfaces of the overlapping subdomains, leading to error propagation and conditioning driven coarse spaces. A main outcome is a fully parallel implementation within the solver class of Code_Aster, assessed on fine discretizations and high levels of parallelism. The approach is validated primarily on unsaturated two-phase, two-component flow configurations in porous media, central to storage- scale thermo-hydro-mechanical simulations. As an additional case of interest, it is also evaluated on large-scale elasto-visco-plasticity simulations, confirming its ability to make simulations more robust and reduce the cost of computations.

Jury

Pr. Victorita DOLEAN	TU Eindhoven	Examiner
Pr. Martin GANDER	Université de Genève	Reviewer
Dr. Luc GIRAUD	Inria	Thesis supervisor
Pr. Pierre GOSSELET	Université de Lille	Examiner
Dr. Carola KRUSE	Cerfacs	Thesis co-supervisor
Pr. Félix KWOK	Université Laval	Reviewer
Dr. Augustin PARRET-FREAUD	Safran	Examiner
Dr. Nicole SPILLANE	CNRS, CMAP	Examiner
Dr. Nicolas TARDIEU	EDF R&D	Invited