A discontinuous Galerkin method for elliptic-hyperbolic equations

We present and analyze a discontinuous Galerkin method for the numerical solution of a class of second-order linear mixed-type partial differential equations, i.e. equations that change their nature from elliptic to hyperbolic through the computational domain. Well-posedness of the discrete problem is established via coercivity in an energy norm, achieved through the Morawetz multiplier technique. We derive $hp$-a priori error estimates in the energy norm, which we use to prove convergence rates for standard and quasi-Trefftz polynomial spaces. Numerical experiments validate the theoretical results.