相关论文: Discrete Exterior Calculus

Discrete exterior calculus offers a coordinate--free discretization of exterior calculus especially suited for computations on meshes over curved manifolds. The discretization of the wedge product, that would be compatible with discrete…

Let $F$ be a discrete Morse function on a simplicial complex $L$. We construct a discrete Morse function $\Delta(F)$ on the barycentric subdivision $\Delta(L)$. The constructed function $\Delta(F)$ "behaves the same way" as $F$, i. e. has…

There are very few results on mixed finite element methods on surfaces. A theory for the study of such methods was given recently by Holst and Stern, using a variational crimes framework in the context of finite element exterior calculus.…

Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of…

The abstract elliptic and parabolic equations on exterior domain are considered. The equations have top-order variable coefficients. The separability properties of boundary value problems for elliptic equation and well-posedness of the…

We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…

We investigate a particular realization of generalized q-differential calculus of exterior forms on a smooth manifold based on the assumption that the N-th power (N>2) of exterior differential is equal to zero. It implies the existence of…

We equip a family of algebras whose noncommutativity is of Lie type with a derivation based differential calculus obtained, upon suitably using both inner and outer derivations, as a reduction of a redundant calculus over the Moyal four…

Vector calculus in three-dimensional space is ubiquitous in applications of mathematics in physics and engineering. Its two-dimensional version is, however, quite rare. Here we try to provide a pedagogical account of the subject. It is…

The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in recent work (collaboration with H.…

Vector fields and line fields, their counterparts without orientations on tangent lines, are familiar objects in the theory of dynamical systems. Among the techniques used in their study, the Morse--Smale decomposition of a (generic) field…

Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes…

We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so…

Skew-symmetric differential forms play an unique role in mathematics and mathematical physics. This relates to the fact that closed exterior skew-symmetric differential forms are invariants. The concept of "Exterior differential forms" was…

We describe discretisations of the shallow water equations on the sphere using the framework of finite element exterior calculus, which are extensions of the mimetic finite difference framework presented in Ringler, Thuburn, Klemp, and…

A Hamiltonian field theory for the macroscopic Maxwell equations with fully general polarization and magnetization is stated in the language of differential forms. The precise procedure for translating the vector calculus formulation into…

In this paper, we develop a geometric, structure-preserving semi-discrete formulation of Maxwell's equations in both three- and two-dimensional settings within the framework of discrete exterior calculus. This approach preserves the…

Basic elements of integral calculus over algebras of iterated differential forms, are presented. In particular, defining complexes for modules of integral forms are described and the corresponding berezinians and complexes of integral forms…

We are interested in differential forms on mixed-dimensional geometries, in the sense of a domain containing sets of $d$-dimensional manifolds, structured hierarchically so that each $d$-dimensional manifold is contained in the boundary of…

We describe a topological predual to differential forms constructed as an inductive limit of a sequence of Banach spaces. This subspace of currents has nice properties, in that Dirac chains and polyhedral chains are dense, and its operator…