Domain decomposition methods are powerful tools to solve real life problems, since they allow to easily devise intrinsically parallel algorithms that can benefit from modern multiprocessors architectures.
This research line focuses on the non conforming version of domain decomposition methods that has the further advantage to allow the use of non matching discretizations and/or heterogeneous methods in different subdomains. For example, refinement can be carried out in each subdomain independently of the neighbors, and it is, in principle, possible to use in each subdomain the best suited method: spectral methods where the solution is expected to be very smooth, finite elements where a complicated geometry requires it, wavelets where isolated singularities in a regular background are expected.
Within the research line, we consider different approaches (mortar method, Nitsche coupling, three-fields formulation, ...) and different type of subdomain discretizations (finite elements both continuous and discontinuous, virtual elements, wavelets, ...), and we deal with different issues. We study the numerical properties of the methods (convergence, stability,..) and we tackle the problem of designing efficient preconditioners (FETI, BDDC, substructuring).
Besides domain decomposition, we also deal, in the research line, with fictitious domain type methods, which also fall in the class of non-conforming methods. In particular we study the theoretical and numerical properties of the Fat Boundary Method.
This is an ongoing collaboration with IRMA-Strasbourg, Université Grenoble-Alpes, MOX-Politecnico di Milano, Politecnico di Torino.
S. Bertoluzza, M. Ismail, B. Maury,
Analysis of the fully discrete Fat Boundary Method, Numer. Math. 118(1), (2011).
S. Bertoluzza, M. Pennacchio, C. Prud'homme, A. Samake
Substructuring preconditioners for h-p Mortar FEM, Tech. Report IMATI-PV n. 10PV14/0/0 (2014)
P. Antonietti, B. Ayuso de Dios, S. Bertoluzza and M. Pennacchio
Substructuring preconditioners for an h-p domain decomposition method with Interior Penalty mortaring, to appear in Calcolo (2014)