Discrete Hodge operators
نویسنده
چکیده
Many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms. Discrete differential forms provide a natural and canonical approach to their discretization. However, much freedom remains concerning the choice of discrete Hodge operators, that is, discrete analogues of constitutive laws. A generic discrete Hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations. We reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus.
منابع مشابه
Polynomial Histopolation, Superconvergent Degrees of Freedom, and Pseudospectral Discrete Hodge Operators
We show that, given a histogram with n bins—possibly non-contiguous or consisting of single points—there exists a unique polynomial histopolant (polynomial of order n with bin averages equal to the histogram’s). We also present histopolating cardinal functions for histograms with contiguous bins. The Hodge star operator from the theory of differential forms is the mapping which puts kand (d − k...
متن کاملAnalysis of Compatible Discrete Operator Schemes - for Elliptic Problems on Polyhedral Meshes
Compatible schemes localize degrees of freedom according to the physical nature of the underlying fields and operate a clear distinction between topological laws and closure relations. For elliptic problems, the cornerstone in the scheme design is the discrete Hodge operator linking gradients to fluxes by means of a dual mesh, while a structure-preserving discretization is employed for the grad...
متن کاملA Discrete Duality Finite Volume Approach to Hodge Decomposition and div-curl Problems on Almost Arbitrary Two-Dimensional Meshes
Abstract. We define discrete differential operators such as grad, div and curl, on general two-dimensional non-orthogonal meshes. These discrete operators verify discrete analogues of usual continuous theorems: discrete Green formulae, discrete Hodge decomposition of vector fields, vector curls have a vanishing divergence and gradients have a vanishing curl. We apply these ideas to discretize d...
متن کاملA field-theoretic model for Hodge theory
We demonstrate that the four (3 + 1)-dimensional (4D) free Abelian 2-form gauge theory presents a tractable field theoretical model for the Hodge theory where the well-defined symmetry transformations correspond to the de Rham cohomological operators of differential geometry. The conserved charges, corresponding to the above continuous symmetry transformations, obey an algebra that is reminisce...
متن کاملVertex-Based Compatible Discrete Operator Schemes on Polyhedral Meshes for Advection-Diffusion Equations
We devise and analyze vertex-based, Péclet-robust, lowest-order schemes for advection-diffusion equations that support polyhedral meshes. The schemes are formulated using Compatible Discrete Operators (CDO), namely primal and dual discrete differential operators, a discrete contraction operator for advection, and a discrete Hodge operator for diffusion. Moreover, discrete boundary operators are...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Numerische Mathematik
دوره 90 شماره
صفحات -
تاریخ انتشار 2001