High-order element techniques such as the Spectral Element Method (SEM) and Discontinuous Galerkin (DG) method have become very popular within the computational fluid dynamics community over the past decade. Both of these techniques utilize the weak form of the a PDE, and their theory is well established. In parallel, a number of element based collocation techniques were developed utilizing the “strong” form of a PDE, such as Spectral Difference, Spectral Volume, and Spectral Multi-domain methods. These collocation type schemes have been unified more recently under the umbrella of “Flux Reconstruction” (FR) methods. Analysis of FR has been limited to a small number of related schemes for which stability and accuracy results have been established. The general development of energy stable FR schemes of a given accuracy remains an open problem.

A unified view of the SEM, DG, and FR techniques will be presented. It will be shown that FR methods can be interpreted as a class of discontinuous Petrov-Galerkin methods utilizing numerical integration. Established convergence and stability results for existing FR schemes are readily explained using this framework. It follows that the future development FR methods with predictable stability and accuracy can draw on the well established theories of the SEM and DG techniques. Concrete examples will be given in the context of the Helmholtz equation.