Spacetime Finite-Element Exterior Calculus and its Applications to Multisymplectic Field Theories

Melvin Leok

Slides are available by following the link.

Finite-element exterior calculus as gained prominence as a method for constructing stable mixed finite-element spaces, however the theoretical development has largely focused on discretizing spatial differential forms. Motivated by variational multisymplectic discretizations of covariant Lagrangian and Hamiltonian field theories, we provide an explicit characterization of the space of Whitney forms and their Hodge dual with respect to a Lorentzian metric. Another notable feature is that this construction does not rely on the use of barycentric coordinates, which results in more efficient computational methods. These low-order finite-element exterior calculus spaces for Lorentzian manifolds are combined with variational multisymplectic integrators to obtain a structure-preserving discretization of Maxwell's equations on a spacetime simplical complex.