Comparison results and unified analysis for first-order finite volume element methods for a Poisson model problem

There exist at least three first-order finite volume element methods with totally different dual meshes and conforming or nonconforming P-1 finite element trial functions and the question arises of whether they are comparable. The fact that the underlying norms are very different does not prevent the proof that the errors are equivalent on any mesh in some norm. This equivalence is independent of the regularity of the exact solution and holds for any coarse or fine mesh with or without local mesh refining, but up to equivalence constants and additional explicit data-oscillation terms of higher order. The equivalence constants depend on the minimal angle of the shape-regular triangulation and the penalization parameter of the discontinuous Galerkin scheme. This also implies quasi-optimality in the sense that the error is bounded by the best approximation of the flux by piecewise constants. An a posteriori error analysis for the discontinuous Galerkin finite volume element scheme is also discussed. The analysis is exemplified for a boundary value model problem for some second-order elliptic partial differential equation in two dimensions.