Related papers: Multisymplecticity in finite element exterior calc…
The EHP and the MCAP provide new rigorous weak variational formalism for a broad range of initial boundary value problems in mathematical physics and mechanics. Both approaches utilize the mixed formulation and lead to the development of…
We study symplectic Laplacians on compact symplectic manifolds with boundary. These Laplacians are associated with symplectic cohomologies of differential forms and can be of fourth-order. We introduce several natural boundary conditions on…
In the present work, we examine and analyze an hp-version interior penalty discontinuous Galerkin finite element method for the numerical approximation of a steady fluid system on computational meshes consisting of polytopic elements on the…
The conformal formulation of the Einstein constraint equations is first reviewed, and we then consider the design, analysis, and implementation of adaptive multilevel finite element-type numerical methods for the resulting coupled nonlinear…
This thesis presents the development of a novel finite element library in Rust based on the principles of Finite Element Exterior Calculus (FEEC). The library solves partial differential equations formulated using differential forms on…
In this paper we propose and investigate a general approach to constructing local energy-preserving algorithms which can be of arbitrarily high order in time for solving Hamiltonian PDEs. This approach is based on the temporal…
We present a new hybrid discrete exterior calculus (DEC) and finite difference (FD) method to simulate fully three-dimensional Boussinesq convection in spherical shells subject to internal heating and basal heating, relevant in the…
Adjoint systems are widely used to inform control, optimization, and design in systems described by ordinary differential equations or differential-algebraic equations. In this paper, we explore the geometric properties and develop methods…
In this paper we discussed the self-adjointness of the Maxwell's equations with variable coefficients $\epsilon$ and $\mu$. Three different Lagrangian are attained. By the Legendre transformation, a multisymplectic Bridge's (Hamilton) form…
Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations,...) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian systems on manifolds equipped with Poisson brackets. The Poisson structure is…
We consider some differential geometric classes of local and nonlocal Poisson and symplectic structures on loop spaces of smooth manifolds which give natural Hamiltonian and multihamiltonian representations for some important nonlinear…
We show how to systematically apply the Faddeev-Jackiw symplectic method to General Relativity (GR) and to GR extensions. This provides a new coherent frame for Hamiltonian analyses of gravitational theories. The emphasis is on the…
Fractional partial differential equations (FDEs) are used to describe phenomena that involve a "non-local" or "long-range" interaction of some kind. Accurate and practical numerical approximation of their solutions is challenging due to the…
A recent paper of Arnold, Falk, and Winther [Bull AMS, 47 (2010)] showed that a large class of mixed finite element methods can be formulated naturally on Hilbert complexes, where using a Galerkin-like approach, one solves a variational…
The fourth-order PDE that models the density variation of smectic A liquid crystals presents unique challenges in its (numerical) analysis beyond more common fourth-order operators, such as the classical biharmonic. While the operator is…
This paper proposes novel decoupled finite element methods for a fourth-order exterior differential equation. Based on differential complexes and the Helmholtz decomposition, the fourth-order exterior differential equation is decomposed…
In this manuscript we present a novel and efficient numerical method for the compressible viscous and resistive MHD equations for all Mach number regimes. The time-integration strategy is a semi-implicit splitting, combined with a hybrid…
Many important physical systems can be described as the evolution of a Hamiltonian system, which has the important property of being conservative, that is, energy is conserved throughout the evolution. Physics Informed Neural Networks and…
In this article we study multisymplectic geometry, i.e., the geometry of manifolds with a non-degenerate, closed differential form. First we describe the transition from Lagrangian to Hamiltonian classical field theories, and then we…
We consider three different methods for the coupling of the finite element method and the boundary element method, the Bielak-MacCamy coupling, the symmetric coupling, and the Johnson-N\'ed\'elec coupling. For each coupling we provide…