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

TL;DR

This work develops a high-order hybrid discretization for Friedrichs systems, enabling robust, general-mesh treatment of diffusion-advection-reaction problems with unknowns on elements and faces. By proving an inf-sup stability result and deriving convergence in $h^{k+1/2}$ (with potential $h^{k+1}$ in reaction-dominated regimes), the authors provide the first complete theory for arbitrary-order hybrid schemes on polytopal meshes and demonstrate static condensation for smaller global systems. The method is validated in three-dimensional scalar and vector DAR settings, as well as a magnetohydrodynamics benchmark, highlighting accurate convergence and physically relevant behavior such as magnetic-field expulsion at higher Reynolds numbers. Overall, the framework offers a provably stable, efficient, and versatile approach for Friedrichs-type models across diffusive and advective regimes, with practical impact for complex multiphysics simulations.

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.

Figures (3)

• Figure 1: Approximation error $|||\underline{I}_h^k{u}-\underline{u}_h|||_{h}$ v. meshsize $h$ for the test described in Section \ref{['subsec:num-scalar_DAR']}. Convergence slopes are displayed for two different mesh families. Specimens of the two mesh families with the smallest diameter are represented on the right.
• Figure 2: Approximation error $|||\underline{I}_h^k{u}-\underline{u}_h|||_{h}$ v. meshsize $h$ for the test described in Section \ref{['subsec:num-vector_DAR']}. Convergence slopes are displayed for the mesh families depicted in Figure \ref{['fig:trend_scalar_DAR']}.
• Figure 3: Numerical results for the test described in Section \ref{['subsec:magn.field.diff.conv']}. The profile of the rotating cylinder is displayed in blue at the center. The numerical solution is shown by displaying the field lines of the magnetic field $B$ stemming from points distributed along the line marked in red. When the cylinder is still ($\mathfrak{Rm}=0$), the magnetic field is equal to the uniform external field $B_0$. When the cylinder moves, the line fields are pushed outside of the cylinder. The expulsion effect is larger for larger values of $\mathfrak{Rm}$.

Theorems & Definitions (18)

• Remark 1: Interpretation of problem \ref{['eq:vector_DAR']}
• Proposition 2: Global discrete integration by parts formula
• proof
• Remark 3: Stabilization for scalar and vector diffusion-advection-reaction
• Remark 4: Static condensation
• Remark 5: Flux formulation
• Remark 6: Comparison with the scheme of Chen.Kang.ea:24
• Proposition 7: Equivalent reformulation of $a_h$
• proof
• Corollary 8: Partial coercivity of $a_h$