Structure-Preserving Hybrid Finite Element Methods

Washington University, Saint Louis MO

Investigators

Abstract

Finite element methods are among the cornerstones of modern scientific computing, combining high performance with strong theoretical guarantees of accuracy and stability. Their applications include several areas of Federal strategic interest: materials and manufacturing, biomedical engineering and biotechnology, structural engineering and civil infrastructure, environmental engineering, and more. They are widely used by scientists and engineers in academia, industry, and national laboratories to simulate large, complex physical systems. However, in order for the simulations produced by these methods to be physically meaningful and trustworthy, it is desirable that they capture certain fundamental physical laws present in the original systems. Certain systems satisfy "conservation laws," which state that some quantity (like mass, energy, or charge) can move through space but not appear or disappear spontaneously, while other systems satisfy "dissipation laws" that require these quantities to grow or decay at a certain rate (e.g., decay of energy due to friction). Although classical finite element methods make it difficult to express these physical laws, the PI has shown that "hybrid" finite element methods provide a way to do so. In this project, the PI will conduct further investigations into hybrid finite element methods, developing and analyzing computational techniques for preserving these important physical structures. The success of this project would lead to new computational methods and improved understanding of current methods for a wide variety of high-value scientific applications, with important ramifications for computational physics and engineering. Many partial differential equations (PDEs), particularly those encountered in physical applications, contain local symmetries, conservation laws, and related structures. At first glance, some of these local structures appear difficult for a finite element method to capture, and consequently, much of the research in this direction has been restricted to finite difference methods on rectangular grids. However, in recent years, the PI has developed a successful research program showing that hybridization provides a route around the apparent obstacles to local-structure-preserving finite elements. This project will investigate several new questions in this direction, organized around two main themes. (1) Local conservation laws with quadratic densities are common in Hamiltonian PDEs, because they arise from linear point symmetries. However, conventional finite element discretization breaks local symmetry, resulting only in global conservation laws. The PI aims to circumvent this obstacle by using hybrid methods, and to extend this to non-conservative systems where a quadratic quantity is dissipated rather than conserved. (2) The PI and his collaborators have recently succeeded in hybridizing finite element exterior calculus (FEEC), which uses differential forms to unify several families of methods for Laplace-type operators. The PI will investigate the application of hybridized FEEC methods to Hamiltonian PDEs, specifically focusing on multisymplectic structure preservation, and to investigate hybrid FEEC methods with strongly (rather than merely weakly) conservative fluxes, including nonconforming and hybridizable discontinuous Galerkin (HDG) methods. 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.