In this project we aim at addressing mathematical and numerical methods based on virtual controls for the coupling of heterogeneous problems described by partial differential equations.
These problems arise in many practical applications. For example, whenever different phenomena have to be taken into account in two or more subregions of the computational domain, or when, for the description and simulation of complex physical phenomena, combinations of hierarchical mathematical models are set up with the aim of reducing the computational complexity.
In both cases, we have to study a multiphysics problem, i.e. a system of heterogeneous problems, where different kind of differential problems are defined in different subregions (either disjoint or overlapping) of the original computational domain.
Virtual control is a powerful technique based on the optimal control theory that has been introduced in domain decomposition method with overlapping subdomains to treat homogeneous problems, either elliptic and parabolic.
The basic idea of this approach consists in introducing suitable functions called ``virtual'' controls which play the role of unknown boundary data on the interfaces of the decomposition and in minimizing in a suitable norm the difference between the solutions of the subproblems on the overlap.
In this project we aim at extending the virtual control method for a class of heterogeneous problems considering both analytical and computational aspects. In particular, we will characterize suitable functionals to be minimized on the overlapping regions in order to ensure both the well-posedness of the problem and a correct description of the physical phenomena of interest. Moreover, we will focus on the development of effective numerical methods and preconditioning techniques for the solution of the optimality systems arising from this approach.
Finally, we will consider several problems of practical interest to validate the methodology that we propose.
Fields of science
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
- natural sciencescomputer and information sciencescomputational sciencemultiphysics
- natural sciencesmathematicsapplied mathematicsnumerical analysis
- natural sciencesmathematicsapplied mathematicsmathematical model
Call for proposal
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