# Advanced Numerical Methods in the Mathematical Sciences

## Yanqiu Wang, Oklahoma State University In Search of Conforming $$H\left(\text{curl}\right)$$ and $$H\left(\text{div}\right)$$ Elements on Polytopes

### Abstract

We construct $$H\left(\text{curl}\right)$$ and $$H\left(\text{div}\right)$$ finite elements on convex polygons and polyhedra. These elements can be viewed as extensions of the lowest order Nedelec-Raviart-Thomas elements, from simplices to general convex polytopes. The construction is based on generalized barycentric coordinates and the Whitney forms. In 3D, it currently requires the faces of the polyhedron be either triangles or parallelograms. Unified formula for computing basis functions are given. The finite elements satisfy discrete de Rham sequences in analogy to the well-known ones on simplices. Moreover, they reproduce existing $$H\left(\text{curl}\right)$$ and $$H\left(\text{div}\right)$$ elements on simplices, parallelograms, parallelepipeds, pyramids, and triangular prisms. Approximation property of the constructed elements is obtained on arbitrary convex polygons and certain polyhedra.