On the computation of the boundary integral of space-time deforming finite elements

We present the integrated-by-parts version of the time discontinuous Galerkin least-squares finite element formulation for the solution of the unsteady compressible Navier-Stokes equations for three dimensional problems involving moving boundaries and interfaces. The deformation of the spatial domain is automatically taken into account by writing the weak form of the problem over its space-time domain. The integration by parts in the three-dimensional spatial case is non-trivial, requiring the application of the Gauss theorem in a 4D space-time continuum. We address the problem by developing an application of the general Stokes' theorem.