Two Hermite analogs of the lowest order Raviart-Thomas mixed element for the convection-diffusion equations

Abstract

Paper presented to the 10th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Florida, 14-16 July 2014.

This work addresses two techniques to solve convection diffusion problems based on Hermite interpolation. More specifcally it deals with an adaptation to the case of these equations of a Hermite nite element method providing flux continuity across inter-element boundaries, shown to be an efficient tool for simulating purely diffusive phenomena[7]. In the latter case the method is the Hermite analog of the celebrated lowest order Raviart-Thomas mixed nite element method known as RT0 [5]. The new methods in turn can be viewed as non trivial improved versions of the RT0 extensions to convection-diffusion problems, in divergence form or not, proposed by Douglas and Roberts [3] more than three decades ago. In contrast to the mixed methods, second order convergence results in the mean square sense are proven to hold for both Hermite nite element approaches, and comparative numerical results illustrate the good performance of the new methods.