2021/03/16 by Brian K. Tran, Tran, Brian, Melvin Leok +1
Mathematics · Physics and Astronomy · Engineering · #Numerical methods for differential equations #Nonlinear Waves and Solitons #Advanced Numerical Methods in Computational Mathematics
paper · pdf · doi:10.48550/arxiv.2103.09338
Compatible discretizations, such as finite element exterior calculus, provide\na discretization framework that respect the cohomological structure of the de\nRham complex, which can be used to systematically construct stable mixed finite\nelement methods. Multisymplectic variational integrators are a class of\ngeometric numerical integrators for Lagrangian and Hamiltonian field theories,\nand they yield methods that preserve the multisymplectic structure and\nmomentum-conservation properties of the continuous system. In this paper, we\ninvestigate the synthesis of these two approaches, by constructing\ndiscretization of the variational principle for Lagrangian field theories\nutilizing structure-preserving finite element projections. In our\ninvestigation, compatible discretization by cochain projections plays a pivotal\nrole in the preservation of the variational structure at the discrete level,\nallowing the discrete variational structure to essentially be the restriction\nof the continuum variational structure to a finite-dimensional subspace. The\npreservation of the variational structure at the discrete level will allow us\nto construct a discrete Cartan form, which encodes the variational structure of\nthe discrete theory, and subsequently, we utilize the discrete Cartan form to\nnaturally state discrete analogues of Noether's theorem and multisymplecticity,\nwhich generalize those introduced in the discrete Lagrangian variational\nframework by Marsden et al. [29]. We will study both covariant spacetime\ndiscretization and canonical spatial semi-discretization, and subsequently\nrelate the two in the case of spacetime tensor product finite element spaces.\n