Hermite finite elements for second order boundary value problems with sharp gradient discontinuities

摘要

In two recent papers the author introduced a finite element method to solve second order elliptic equations in N-dimensional space, for N=2 and N=3 respectively, providing flux continuity across inter-element boundaries on the basis of Hermite interpolation in an N-simplex. After defining this method in the framework of diffusion-like problems with anisotropic diffusion tensors, another N-simplex based Hermite finite element method to solve the same class of problems is considered. The latter can be viewed as a variant of the popular lowest-order Raviart–Thomas mixed element known as RT0. A convergence analysis of this method is given, showing that, in contrast to RT0, it is second order accurate in L2. Some numerical examples comparing the three methods are given.