Space-time virtual elements for the heat equation

We propose and analyze a space-time virtual element method for the discretization of the heat equation in a space-time cylinder, based on a standard Petrov-Galerkin formulation. Local discrete functions are solutions to a heat equation problem with polynomial data. Global virtual element spaces are nonconforming in space, so that the analysis and the design of the method are independent of the spatial dimension. The information between time slabs is transmitted by means of upwind terms involving polynomial projections of the discrete functions. We prove well posedness and optimal error estimates for the scheme, and validate them with several numerical tests.