Development and application of high-order discontinuous CVFEM algorithms

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.