Direct Finite Elements on Convex Polygons and Polyhedra

Mathematical modeling and computational simulation is a safe and cost-effective way to understand and test natural and engineered systems. Often one needs to determine the rate of change of some quantity of interest. In that case, the models will generally include partial differential equations (PDEs), which can be solved computationally using finite element methods.

The quantity of interest is a function of, say, space that is approximated over small simple shapes called elements which together form a computational mesh. Meshes are often composed of simple triangles and rectangles (or simplices and bricks in 3D), but these meshes have various limitations. Meshes of polygons (or polyhedra in 3D) are more flexible, but there are not many finite elements available for these meshes that accurately approximate the function.

This project will develop practical finite elements on polygons and simple polytopes that merge together continuously and are provably accurate. The new finite elements are expected to have an impact on broad areas of science and engineering by making the use of polytopal meshes more accessible. They may also have a broader impact in terms of interpolation and visualization of functions in computer graphics and in the representation of data.

Specific applications to the geosciences (subsurface modeling for energy and water resource management) are planned. The project will support the research of students who will work in an interdisciplinary environment. Students thus trained are in high demand in industrial and governmental labs, as well as in academia.

Many computational scientists are interested in defining finite elements on nonstandard, polytopal elements (polygons and polyhedra). Often one desires a conforming approximation with an explicit finite element basis. The latter is particularly helpful when dealing with nonlinear PDEs and coupled systems of equations.

One could define a finite element on a reference element and map it to the physical element. However, there is a loss in accuracy due to the fact that the map is non-affine, and also mapped mixed elements generally do not preserve the divergence free property in a pointwise sense. The proposed research involves explicit and practical construction of minimal degree of freedom finite elements on polygons and simple polytopes, as well as the mathematical analysis of their approximation properties and numerical applications.

The approach is to define H1-conforming shape functions in terms of polynomials posed directly on the physical element and supplement the space with a small number of explicitly defined nonpolynomial functions. These supplemental functions will be defined as rational functions that allow us to define a nodal basis. The de Rham theory can then be used to define mixed finite elements.

The objectives are to: 1) Develop direct serendipity and mixed finite elements on 2D convex polygons; 2) Develop direct serendipity and vector-valued H(curl) and H(div) finite elements on 3D cuboidal hexahedra; 3) Develop direct serendipity and vector-valued finite elements on general 3D convex polyhedra (however, it is not expected that this objective will be fully resolved within the three year duration of the project); and 4) Use the new finite elements to solve problems in subsurface flow applications, including some that can use and explore hp-refinement properties of the polygonal elements.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.