Novel Raviart–Thomas Basis Functions on Anisotropic Finite Elements

Abstract Recently, H ⁢ ( div ) \mathbf{H}(\mathrm{div}) -conforming finite element families were proven to be successful on anisotropic meshes, with the help of suitable interpolation error estimates. In order to ensure corresponding large-scale computation, this contribution provides novel Raviart–Thomas basis functions, robust regarding the anisotropy of a given triangulation. This new set of basis functions on simplices uses a hierarchical approach, and the orientation of the basis functions is inherited from the lowest-order case. In the higher-order case, the new basis functions can be written as a combination of the lowest-order Raviart–Thomas elements and higher-order Lagrange-elements. This ensures robustness regarding the mesh anisotropy and assembling strategies as demonstrated in the numerical experiments.