A current problem that I am working on requires me to compute the solution from the heat diffusion evolution on a discontinuous function. More precisely - I have a Delaunay triangulation and within each triangle a polynomial function is defined such that it can be discontinuous at the edges. For a constant function within each triangle the two-point flux approximation scheme is applicable (precisely because I have a Delaunay triangulation).
Is there a DG extension/analogue to such a cell-centered scheme? I would like to be able to use a similar cell-centered scheme on cells with higher polynomial degrees.
To put it simply, I am looking for a (simpler) method to perform homogeneous diffusion on a mesh where discontinuous polynomial functions are defined within each cell (preferably one that extends to higher degree polynomials unlike the FVM scheme that I mentioned).