The CAVE project is collocated in the area of computational/applied mathematics with a strong connection with engineering. A mathematical model is a description of some phenomena of interest (in turn associated to some application such as weather forecast, design of an aircraft, etc..) in terms of mathematical functions and "equations". The majority of models cannot be solved by pen and paper, and thus in order to be useful need to be discretized (approximated by a numerical method and translated into a form that is edible for a computer) and solved computationally. Mathematics has a key role in designing the numerical method and guaranteeing theoretically its effectiveness. Since the majority of problems are set in some space domain (such as a piece of a vessel if one is studying blood flow, or a building if one is investigating earthquake effects, etc..) a key step in essentially all methods is some kind of discretization (decomposition) of such domain (called mesh).
The project CAVE is focused on developing an innovative numerical method for the discretization of the above models, the Virtual Element Method, and thus carrying an impact on the related applications. The novel side with respect to the pre-existing technology is related to many aspects, the first one being a much larger freedom in building the mesh, as the scheme can use any general polyhedral decomposition of the domain. Other aspects are, for instance, the capability to realize exactly certain physical conditions of the solution (such as being incompressible, or needing a higher regularity). The project has a wide theoretical foundation to be explored, in addition to practical computations and applications (such as in structural mechanics and electro-cardiology). The final goal, in a wide sense, is to develop, analyze and deliver a new method that, in many problems/situations, can yield much better (more accurate, more efficient,..) results with respect to the existing ones.
The project CAVE was successful in its main objectives, by actually developing novel discretization approaches for a wide range of problems, going from solid and fluid mechanics to electro-magnetism, that offer novel advantages with respect to existing competitors. Such developments also include (and where boosted by) a full theoretical background, plus implementation of computer codes and numerical benchmark testing. Furthermore, the Virtual Element Method enjoyed a strong growth in terms of attention and community size, both in the mathematical and in the engineering areas. As expected at the start of the action, it is still too early for embedding into the industry.
