Virtual Element based Model Order Reduction

1) Wider research context
We focus on the numerical approximation of solutions of parametric Partial Differential Equations (PDEs). The two approximation techniques that we will exploit are the Virtual Element Method and the Reduced Basis Method. The Virtual Element Method is a spatial discretization technique that has many advantages: among the others, we mention the possibility to deal with polygonal meshes, with curvilinear boundaries and interfaces, and the consequent suitability for adaptive methods. The Reduced Basis Method is a model order reduction technique, which is of great interest for the applied mathematics community nowadays, given its ability to exploit high performance computing.

2) Objectives
The goal of the proposed research is to bridge the Virtual Element Method with the Reduced Basis Method, with the aim of obtaining a new, powerful numerical approximation tool for parametric Partial Differential Equations.
We aim at developing and analyzing computational models and techniques for a model order reduction that is based on the Virtual Element discretization.
We are then interested in applying and testing this approximation methodology to advanced problems, such as Fluid-Structure Interaction problems. We will focus on monolithic as well as segregated approaches: we will formulate and analyze suitable algorithms, based on the properties of the underlying Virtual Element discretization.

3) Approach
To reach the proposed goals, we will develop model order reduction packages within an already existing, freeware and available upon request C++ library, that has been specifically designed for the Virtual Element Method. We will formulate and analyze suitable algorithms and techniques, in order to fit them within the Virtual Element framework. We exploit these new computational tools on different applications of interest: starting from simple, linear, parametric problems, the goal is to apply and test the new methodology on Fluid-Structure Interaction problems.
To address these complex problems, we start from a steady, linear, geometrically parametrized problem, which will allow us to formulate and analyze different approaches; we then move progressively forward to unsteady, fully nonlinear coupled problems.

4) Level of originality
The merging between the Virtual Element Method and the Reduced Basis Method, and the design of a Virtual Element based model order reduction library, represents a fundamental scientific contribution, that has never been studied, even at a preliminary stage. The outcomes of this project will give rise to methodological results and computational tools, from which the applied mathematics community can certainly benefit.

5) Primary researchers involved
The primary researchers involved in the project are: Monica Nonino as the Principal Investigator, Ilaria Perugia as the mentor, Franco Dassi and Thomas Wick as external collaborators.