Hybrid Finite Element Methods for Geometric Partial Differential Equations

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Finite element methods are among the most important, powerful tools in scientific computing. 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. These physical systems often obey "conservation laws," which state that some quantity -- like mass, or energy, or electric charge -- can move around in space, but cannot appear or disappear spontaneously. It is desirable that simulations of these systems also obey these conservation laws, because they are so fundamental; otherwise, the simulated results may not be physically meaningful or trustworthy. However, this is not always the case with current methods. In this project, the PI will develop and analyze finite element methods that obey these conservation laws and preserve related physical properties. The success of this project would lead to new computational methods and improved understanding of current methods for a wide variety of scientific applications. Because the specific applications addressed by the proposed research are of high scientific value, this could have important ramifications for computational physics and engineering.The PI proposes to investigate structure-preserving hybrid finite element methods for partial differential equations (PDEs) containing local symmetries, invariants, and conservation laws. In applications, these locally-invariant geometric structures often have important physical meaning (e.g., conservation of charge in electromagnetics), so it is desirable to devise conservative numerical methods that preserve these structures exactly rather than approximately. Hybrid methods provide a natural framework for this, since one may examine local invariants, element-by-element, in terms of numerical traces and fluxes on their boundaries. The proposed research consists of two main components. (1) Hamiltonian PDEs, which are ubiquitous in physical applications, satisfy the multisymplectic conservation law, which is closely tied to physically-important reciprocity phenomena, traveling waves, dispersion relations, and bifurcations. The PI will extend his recent joint work on multisymplectic hybridizable discontinuous Galerkin (HDG) methods for spatial discretization to time-evolution problems, using spatial HDG semidiscretization and spacetime HDG methods. (2) While the time evolution of Maxwell's equations automatically preserves a divergence constraint associated with charge conservation, this is generally not true for finite element discretizations. Preliminary results show that this can be resolved using a class of hybrid methods, where charge conservation holds in the sense of the numerical electric flux. The PI proposes to extend this by analyzing nonconforming hybrid methods for the Maxwell eigenvalue problem, as well as hybrid methods for Yang-Mills theory using finite element exterior calculus.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.