AbstractA finite element cochain complex on Cartesian meshes of any dimension based on the $$H^1$$
H
1
-inner product is introduced. It yields $$H^1$$
H
1
-conforming finite element spaces with exterior derivatives in $$H^1$$
H
1
. We use a tensor product construction to obtain $$L^2$$
L
2
-stable projectors into these spaces which commute with the exterior derivative. The finite element complex is generalized to a family of arbitrary order.
In this paper we will provide a new operatorial counterpart of the path-integral formalism of classical mechanics developed in recent years. We call it new because the Jacobi fields and forms will be realized via finite dimensional matrices. As a byproduct of this we will prove that all the operations of the Cartan calculus, such as the exterior derivative, the interior contraction with a vector field, the Lie derivative and so on, can be realized by means of suitable tensor products of Pauli and identity matrices.
Given a cellular complex, we construct spaces of differential forms which form a complex under the exterior derivative, which is isomorphic to the cochain complex of the cellular complex. The construction applies in particular to subsets of Euclidean space divided into polyhedra, for which it provides, for each k, a space of k-forms with a basis indexed by the set of k-dimensional cells. In the framework of mimetic finite differences, the construction provides a conforming reconstruction operator. The construction requires auxiliary spaces of differential forms on each cell, for which we provide two examples. When the cells are simplexes, the construction can be used to recover the standard mixed finite element spaces also called Whitney forms. We can also recover the dual finite elements previously constructed by A. Buffa and the author on the barycentric refinement of a two-dimensional mesh.
Many integrable hierarchies of differential equations allow a variational
description, called a Lagrangian multiform or a pluri-Lagrangian structure. The
fundamental object in this theory is not a Lagrange function but a differential
$d$-form that is integrated over arbitrary $d$-dimensional submanifolds. All
such action integrals must be stationary for a field to be a solution to the
pluri-Lagrangian problem. In this paper we present a procedure to obtain
Hamiltonian structures from the pluri-Lagrangian formulation of an integrable
hierarchy of PDEs. As a prelude, we review a similar procedure for integrable
ODEs. We show that exterior derivative of the Lagrangian $d$-form is closely
related to the Poisson brackets between the corresponding Hamilton functions.
In the ODE (Lagrangian 1-form) case we discuss as examples the Toda hierarchy
and the Kepler problem. As examples for the PDE (Lagrangian 2-form) case we
present the potential and Schwarzian Korteweg-de Vries hierarchies, as well as
the Boussinesq hierarchy.
This chapter begins by examining p-forms and the exterior product, as well as the dual of a p-form. Meanwhile, the exterior derivative is an operator, denoted d, which acts on a p-form to give a (p + 1)-form. It possesses the following defining properties: if f is a 0-form, df(t) = t f (where t is a vector of Eₙ), which coincides with the definition of differential 1-forms. Moreover, d(α + β) = dα + dβ, where α and β are forms of the same degree. Moreover, the exterior calculus can be used to obtain a compact and elegant formulation of Maxwell’s equations.
$$H^1$$-conforming finite element cochain complexes and commuting quasi-interpolation operators on Cartesian meshes
