Space–time virtual elements for the heat equation
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We present a space–time virtual element method for parabolic problems, based on a standard Petrov–Galerkin weak formulation. The trial and test spaces consist of functions that solve local time-dependent PDEs with polynomial data and satisfy a Crouzeix–Raviart-type condition across time-like facets. This construction allows for a unified analysis in any spatial dimension. The information between time slabs is transmitted by means of upwind terms involving polynomial projections of the discrete functions. After establishing a priori error estimates, we validate them on some numerical examples and compare the results with those obtained for conforming space–time finite elements. In addition, we introduce a residual-type error estimator to demonstrate the flexibility of the method in supporting adaptive refinements. In particular, we verify its reliability and efficiency for h-adaptive strategies, compare the performance of the scheme with that of conforming finite element methods, and investigate the quasi-efficiency of the error estimator for p- and hp-refinements.