Finite Element Exterior Calculus with Smoother Piecewise Polynomials

Brown University, Providence RI

Investigators

Abstract

Finite element methods are the computational workhorse in simulating many problems in engineering, physics, chemistry and biology. For each particular application, different type of finite elements are needed. In 1980, Nedelec made a monumental connection between different finite elements that has found even more applications. This project will generalize these connections to different finite elements of smoother type. These connections will allow us to tackle new applications. The research will build and analyze finite element spaces on simplicial meshes in arbitrary dimension that fit in a finite element complex following the framework of the finite element exterior calculus (FEEC). The distinctive feature of this proposal is that we will build spaces that are smoother than traditional spaces (e.g. Whitney/Nedelec forms). Smoother spaces are more natural for some applications: plate problems, fluid flow problems. We will accomplish this by using splits of simplices that provide more structure than an arbitrary simplicial decomposition of a domain. The left most spaces of the complex will coincide with functions spaces that have been studied in the spline community. The far right spaces are ones that are associated with inf-sup stable finite element spaces for fluid flow problems. Thus, we plan to connect these function spaces in a natural way into a finite element complex. In addition, we will explore connections with the spline and applied algebraic geometry communities. 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.