The discontinuous control-volume finite-element method is applied to the one-dimensional advection-diffusion equation and
validated on relevant test cases. The technique merges the features of the classical finite-volume method, as robustness and local
conservation properties [1], with those of the discontinuous Galerkin finite-element method, known for the capability of handling
large gradients or discontinuities with high accuracy [2]. On the other hand, most finite-volume methods attain relatively low
orders of spatial accuracy and resolution characteristics, particularly on unstructured meshes. To achieve high-order accuracy,
the proposed technique adopts polynomial shape functions of any degree as in spectral finite-element methods [3]. In many
applications high resolution is not needed in the whole domain, which results also in a loss of computational resources. We thus
apply an automatic p-refinement technique which adapts the polynomial order at element level, according to the local behavior
of the computed solution. Element-wise p-adaption can be easily achieved with discontinuous Galerkin methods, where the
inter-element continuity is imposed in weak form.
