Thesis Summary:

In this work we propose and analyze an HDG method for the Stokes equation whose domain is discretized by two independent polygonal subdomains with different meshsizes. This causes a non-conformity at the intersection of the subdomains or might even leave a gap (unmeshed region) between them. In order to appropiately couple these two different discretizations, we propose suitable transmission conditions such that we preserve the high order of the scheme. On the other hand, stability estimates are established in order to show the well-posedness of the method and the error estimates. In particular, for smooth enough solutions, the L2 norm of the errors associated to the approximations of the velocity gradient, the velocity and the pressure are of order hk+1, where h is the meshsize and k is the polynomial degree of the local approximation spaces. Moreover, the method presents superconvergence of the velocity trace and the divergence-free postprocessed velocity. Finally, we show numerical experiments that validate our theory and the capacities of the method.