Hermite finite elements for diffusion phenomena

Two new Hermite finite elements are shown to be an advantageous alternative to well-known mixed methods in the simulation of diffusion processes in heterogeneous anisotropic media. Both are N-simplex based for N=2 and N=3 and provide flux continuity across inter-element boundaries. One of the methods denoted by P2H was introduced by the first author and collaborator for the case of homogeneous and isotropic media. Its extension to the case of heterogeneous and/or anisotropic cases is exploited here, keeping an implementation cost close to the popular Raviart-Thomas mixed finite element of the lowest order, known as RT0. The other method studied in detail in this work is a new Hermite version of the latter element denoted by RT0M. Formal results are given stating that, at least in the case of a constant diffusion, RT0M is significantly more accurate than RT0, although both elements have essentially the same implementation cost. A thorough comparative numerical study of the Hermite methods and RT0 is carried out in the framework of highly heterogeneous media among other cases. It turns out that both are globally superior all the way, and roughly equivalent to each other in most cases.