A new high-order finite volume element method with spectral-like resolution

In this work, a new high-order finite volume element method with good spatial resolution characteristics is presented. The method is based on a functional representation of the unknowns based on the finite element method, a balance of physical quantities in weak formulation obtained by using the finite volume method, and an implicit reduction of some of the unknowns obtained by enforcing functional relations between some of them. Applications to hyperbolic and elliptic operators in 1D and 2D, as well as to Navier-Stokes equations for incompressible flows are presented.