A Novel Implementation of Hermite Finite Element Methods in the Deal.ii Library

A high-order finite element method based on Hermite interpolation polynomials is described and a software implementation in the deal.II finite element library (https://dealii.org) is proposed. This method can be used to strongly enforce continuity of spatial derivatives as well as solution value over element boundaries, which is necessary for solving fourth order partial differential equations (PDEs) in primal form and useful for some second order PDEs. The implementation is verified by solving the wave equation based on a modification of the PDE with a Lax-Wendroff-like procedure, allowing a high-order solution method with negligible additional computing time.