Collaborative Research: Transforming Serendipity Elements from Theory to Practice

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This project aims to significantly reduce the computation time and effort required to carry out a wide variety of simulations used in modern scientific and engineering applications. In settings as varied as magnetohydrodynamics (used in studies of nuclear fusion) and electro-diffusion (used in studies of cardiac arrhythmia), a technique called the "finite element method" is a preferred computational procedure for designing and computing highly accurate descriptions of the relevant physical phenomena. While finite element methods have been in use for nearly half a century, recent mathematical insights have indicated that certain kinds of methods could be implemented in such a way that they would produce results of the same order of accuracy while requiring orders of magnitude less computational work. Such techniques are known as "serendipity methods" and serve as the focus of the proposed work. A key impact of the research will be incorporation of these methods into a widely used, open source, community-developed software package known as the Firedrake Project.From high-level method analysis to specific computational tricks, the proposed work will significantly expand understanding of the benefits and limitations of serendipity finite element methodologies. For serendipity elements, the project will make precise the common mantra of "same accuracy, less work" by simultaneously exploring the expected computational benefits via algebraic and analytical techniques, as well as actual computational benefits via implementation and numerical simulations within the Firedrake software package. On the theoretical side, the research will advance knowledge via investigation of the algebraic structure of serendipity spaces and its exploitation to allow new Helmholtz decompositions of the associated polynomial spaces, identification of efficient "short injections" of serendipity bases into standard tensor product bases, and development of sparse quadrature rules. The investigators will combine expertise on element mapping and physically-defined basis elements to create an order-preserving handling of non-affinely mapped square element geometries, a key roadblock in prior attempts at implementation of serendipity 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.