Finite element methods encompass a broad and popular class of algorithms for approximately solving partial differential equations (PDE). A variational form of the PDE is discretized by posing the variational problem in finite dimensional spaces, thereby obtaining a large (sparse) linear system whose solution enables the construction of an approximate solution to the PDE. The quality of this approximation is clearly dependent on the choice of finite dimensional space, and finite element methods construct these spaces by first partitioning the (bounded) domain into mesh cells, and then defining local finite dimensional function spaces on each cell---typically in terms of polynomials. Some notion of continuity (related to the underlying PDE) of these functions across cell interfaces is enforced either explicitly or by some penalization method.
Traditionally, mesh cells have been restricted to a relatively small class of shapes: for example, triangles and quadrilaterals in 2D, and tetrahedra, hexahedra and triangular prisms in 3D. Over the past several years, however, there has been growing interest in the development of finite element methods on meshes whose cells are allowed to be general polytopes. Natural questions arise:
- How should the local spaces be defined?
- What are the relevant approximation properties of these local and global spaces?
- How can we efficiently compute using these spaces?
- Why even bother with such schemes?
After briefly considering some of the motivations for considering more exotic meshes, and providing an overview of several different approaches currently being developed, we discuss our contributions in this direction. A unique feature of our work is that we allow for even more general mesh cells, e.g. curvilinear polygons, and we will highlight some of the additional challenges that this generalization poses. Numerical examples will be provided that illustrate many of the features of this approach.