Conforming Harmonic Finite Elements on the Hsieh-Clough-Tocher Split of a Triangle

Abstract

We construct a family of conforming piecewise harmonic finite elements on triangulations. Because the dimension of harmonic polynomial spaces of degree ≤ $k$ is much smaller than the one of the full polynomial space, the triangles in the partition must be refined in order to achieve optimal order of approximation power. We use the Hsieh-Clough-Tocher split: the barycenter of each original triangle is connected to its three vertices. Depending on the polynomial degree $k$, the original triangles have some minor restrictions which can be easily fulfilled by small perturbations of some vertices of the original triangulation. The optimal order of convergence is proved for the conforming harmonic finite elements, and confirmed by numerical computations. Numerical comparisons with the standard finite elements are presented, showing advantages and disadvantages of the harmonic finite element method.