Space--time Morawetz formulations for the wave equation

Most numerical schemes for the approximations of initial--boundary value problem (IBVP) for evolution PDEs rely on separate discretisations of the space and time variables, i.e. either the method of lines or Rothe's method. Space--time methods, instead, consists of simultaneous discretisation in both variables. Even if they require the solution of large algebraic systems, space--time methods can be advantageous in terms of local mesh refinement, adaptivity, parallelisation and treatment of moving interfaces and boundaries. For linear transient wave problems, most numerical schemes belong to the discontinuous-Galerkin (DG) family. Fewer conforming Galerkin methods have been proposed and almost all of them either rely on strong assumptions on the discrete space or on the space--time mesh, in terms of a CFL condition. Ideally, one would like a formulation that is unconditionally stable for a wide array of discrete spaces, on generic polyhedral space--time meshes. We propose two variational formulations for the wave equation with impedance and Dirichlet boundary conditions that are derived by multiplying the PDE with a special test function called the Morawetz multiplier and integrating by parts. We first present a conforming formulation that is well-posed on star-shaped impedance cavities and star-shaped Dirichlet obstacles. Coercivity and continuity are proved in a norm stronger than space--time $H^1$, with elementary vector calculus tools and explicit parameters. $H^2$-conforming finite element methods can be used to stably discretise the formulation. Furthermore, we derive an a-posteriori error estimator that is reliable and efficient, with explicit multiplicative constants, and we employ it in an adaptive numerical scheme based on hierarchical splines. The second formulation we propose is a C0-interior penalty discretisation of the first variational formulation introduced. Such formulation has the advantage that simpler finite element spaces (e.g. $H^1$-conforming) and arbitrary space--time meshes can be used. To prove well-posedness we introduce trace and inverse inequalities that are dimensionally consistent. Furthermore, we derive an a-posteriori error estimator for which reliability and efficiency are proved.