Nonconforming space-time methods for evolution PDEs

Even though the foundation of space–time Galerkin methods dates back to the 70s of the last century, and a few contributions were developed in the twenty years to follow, in the last two decades there has been a growing attention on this topic. In this dissertation, space–time Galerkin methods are designed for the discretization of the Schrödinger and heat equations. For the Schrödinger equation, we design an ultra-weak space–time discontinuous Galerkin variational formulation on general prismatic meshes. The method is well-posed, stable and quasi-optimal for very general discrete spaces. The approximation properties of the method are studied for four choices of discrete spaces: i) the polynomial Trefftz space for problems with zero potential; ii) a pseudo-plane wave Trefftz space for problems with piecewise-constant potentials; iii) the full polynomial space, and iv) a quasi-Trefftz polynomial space for piecewise-smooth potentials. For the heat equation, we design a nonconforming space–time virtual element method, which is also the first space–time virtual element discretization of a time-dependent PDE in the literature. The method is proven to be well-posed, based on a discrete inf-sup condition. An a priori error analysis show that optimal h-convergence rates are obtained for sufficiently smooth solutions. The method performs well also for singular solutions such as those arising from the incompatibility of initial and boundary conditions. Since the method allows for nonmatching space-like and time-like facets, it is naturally suitable for h- and hp-versions.