Hybrid Methods for Friedrichs Systems with Application to Scalar and Vector Diffusion-Advection Problems

In this work we study arbitrary-order hybrid discretizations of Friedrichs systems. Friedrichs systems provide a framework that goes beyond the standard classification of partial differential equations into hyperbolic or elliptic, and are thus particularly suited for problems that include both diffusive and advective terms. The family of numerical schemes proposed in this work hinge on hybrid spaces with unknowns located at elements and faces. They support general meshes, are locally conservative and, compared with traditional Discontinuous Galerkin discretizations, lead to smaller algebraic systems once static condensation has been applied. We carry out a complete stability and convergence analysis, which appears to be the first of its kind. The performance of the method is illustrated on scalar and vector three-dimensional diffusion-advection-reaction problems.