Large scale nonconforming domain decomposition methods

Keywords :

• mortar finite element method
• substructuring preconditioner
• high-performance computing
• linear algebra
• Feel++
• décomposition de domaine
• méthode des éléments finis joints
• préconditionneur par sousstructuration
• calcul haute performance

This thesis investigates domain decomposition methods, commonly classified as either overlapping Schwarz methods or iterative substructuring methods relying on nonoverlapping subdomains. We mainly focus on the mortar finite element method, a nonconforming approach of substructuring method involving weak continuity constraints on the approximation space. We introduce a finite element framework for the design and the analysis of the substructuring preconditioners for an efficient solution of the linear system arising from such a discretization method. Particular consideration is given to the construction of the coarse grid preconditioner, specifically the main variant proposed in this work, using a Discontinuous Galerkin interior penalty method as coarse problem. Other domain decomposition methods, such as Schwarz methods and the so-called three-field method are surveyed with the purpose of establishing a generic teaching and research programming environment for a wide range of these methods. We develop an advanced computational framework dedicated to the parallel implementation of numerical methods and preconditioners introduced in this thesis. The efficiency and the scalability of the preconditioners, and the performance of parallel algorithms are illustrated by numerical experiments performed on large scale parallel architectures.

Directors:

• Mr Christophe Prud'homme (Professeur - Université de Strasbourg )
• Mr Christophe Picard (Maître de Conférence - Grenoble INP )

Raporteurs:

• Mr Damien Tromeur Dervout (Professeur - Université Lyon 1 )
• Mr Zakaria Belhachmi (Professeur - Université Haute Alsace )

Examinators:

• Mr Frédéric Nataf (Directeur de recherche - Université Pierre et Marie Curie )
• Mr Eric Blayo-Nogret (Professeur - Université Joseph Fourier-Université Grenoble Alpes )