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

TL;DR

This paper introduces a new class of hybrid discretization methods for Friedrichs systems, capable of handling complex PDEs with diffusive and advective components, demonstrating stability, convergence, and efficiency in 3D problems.

Contribution

It develops arbitrary-order hybrid schemes for Friedrichs systems with stability and convergence analysis, supporting general meshes and reducing algebraic system size compared to traditional methods.

Findings

Methods are stable and convergent for scalar and vector problems.

Support for general meshes enhances flexibility.

Reduced algebraic systems improve computational efficiency.

Abstract

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.