This paper details the techniques and algorithms implemented in Kahler, a Python library that implements discrete exterior calculus on arbitrary Hermitian manifolds. Borrowing techniques and ideas first implemented in PyDEC, Kahler provides a uniquely general framework for computation using discrete exterior calculus. Manifolds can have arbitrary dimension, topology, bilinear Hermitian metrics, and embedding dimension. Kahler comes equipped with tools for generating triangular meshes in arbitrary dimensions with arbitrary topology. Kahler can also generate discrete sharp operators and implement de Rham maps. Computationally intensive tasks are automatically parallelized over the number of cores detected. The program itself is written in Cython--a superset of the Python language that is translated to C and compiled for extra speed. Kahler is applied to several example problems: normal modes of a vibrating membrane, electromagnetic resonance in a cavity, the quantum harmonic oscillator, and the Dirac-Kahler equation. Convergence is demonstrated on random meshes.
We derive a numerical method for Darcy flow, hence also for Poisson's equation in mixed (first order) form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is one of its discretizations on simplicial complexes such as triangle and tetrahedral meshes. DEC is a coordinate invariant discretization, in that it does not depend on the embedding of the simplices or the whole mesh. We start by rewriting the governing equations of Darcy flow using the language of exterior calculus. This yields a formulation in terms of flux…
We are interested in the numerical reconstruction of a vector field with prescribed divergence and curl in a general domain of R 3 or R 2 , not necessarily contractible. To this aim, we introduce some basic concepts of finite element exterieur calculus and rely heavily on recent results of P. Leopardi and A. Stern. The goal of the paper is to take advantage of the links between usual vector calculus and exterior calculus and show the interest of the exterior calculus framework, without too much prior knowledge of the subject. We start by describing the method used for contractible domains and …
TRiSK-type numerical schemes are widely used in both atmospheric and oceanic dynamical cores, due to their discrete analogues of important properties such as energy conservation and steady geostrophic modes. In this work, we show that these numerical methods are best understood as a discrete exterior calculus (DEC) scheme applied to a Hamiltonian formulation of the rotating shallow water equations based on split exterior calculus. This comprehensive description of the differential geometric underpinnings of TRiSK-type schemes completes the one started in \cite{Thuburn2012,Eldred2017}, and prov…
We give a systematic and self-contained account of the construction of geometrically decomposed bases and degrees of freedom in finite element exterior calculus. In particular, we elaborate upon a previously overlooked basis for one of the families of finite element spaces, which is of interest for implementations. Moreover, we give details for the construction of isomorphisms and duality pairings between finite element spaces. These structural results show, for example, how to transfer linear dependencies between canonical spanning sets, or give a new derivation of the degrees of freedom.
In this paper, we present several new a posteriori error estimators and two adaptive mixed finite element methods \textsf{AMFEM1} and \textsf{AMFEM2} for the Hodge Laplacian problem in finite element exterior calculus. We prove that \textsf{AMFEM1} and \textsf{AMFEM2} are both convergent starting from any initial coarse mesh. A suitably defined quasi error is crucial to the convergence analysis. In addition, we prove the optimality of \textsf{AMFEM2}. The main technical contribution is a localized discrete upper bound. As opposed to existing literature, our results work on Lipschitz domains wi…
This paper proposes a finite element method for solving the periodic steady-state problem for the scalar-valued and vector-valued Poisson equations, a simple reduction model of the Maxwell equations under the Coulomb gauge. Introducing a new potential variable, we reformulate two systems composed of the scalar-valued and vector-valued Poisson problems to a single Hodge-Laplace problem for the 1-form in $\mathbb{R}^4$ using the standard de Rham complex. Consequently, we can directly apply the Finite Element Exterior Calculus (FEEC) theory in $\mathbb{R}^4$ to deduce the well-posedness, stabilit…
Discrete exterior calculus (DEC) for the surface Navier-Stokes equation
Institute of Scientific Computing, TU Dresden（德累斯顿工业大学科学计算研究所）；Dresden Center for Computational Materials Science (DCMS), TU Dresden（德累斯顿工业大学德累斯顿计算材料科学中心）；Center for Systems Biology Dresden (CSBD)（德累斯顿系统生物学中心）
The problem of solving partial differential equations (PDEs) on manifolds can be considered to be one of the most general problem formulations encountered in computational multi-physics. The required covariant forms of balance laws as well as the corresponding covariant forms of the constitutive closing relations are naturally expressed using the bundle-valued exterior calculus of differential forms or related algebraic concepts. It can be argued that the appropriate solution method to such PDE problems is given by the finite element exterior calculus (FEEC). The aim of this essay is the expos…
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 dense matrices arising from standard discretization procedures. In this paper, we begin to extend the well-established computational toolkit of Discrete Exterior Calculus (DEC) to the fractional setting, focusing on proper discretization of the fractional derivative. We define a Caputo-like fractional discrete derivative, in terms of the standard discrete e…
We introduce $Ψ\mathrm{ec}$, a local spectral exterior calculus for the two-sphere $S^2$. $Ψ\mathrm{ec}$ provides a discretization of Cartan's exterior calculus on $S^2$ formed by spherical differential $r$-form wavelets. These are well localized in space and frequency and provide (Stevenson) frames for the homogeneous Sobolev spaces $\dot{H}^{-r+1}( Ω_ν^{r} , S^2 )$ of differential $r$-forms. At the same time, they satisfy important properties of the exterior calculus, such as the de Rahm complex and the Hodge-Helmholtz decomposition. Through this, $Ψ\mathrm{ec}$ is tailored towards structure…
Arnold, Falk, & Winther, in "Finite element exterior calculus, homological techniques, and applications" (2006), show how to geometrically decompose the full and trimmed polynomial spaces on simplicial elements into direct sums of trace-free subspaces and in "Geometric decompositions and local bases for finite element differential forms" (2009) the same authors give direct constructions of extension operators for the same spaces. The two families -- full and trimmed -- are treated separately, using differently defined isomorphisms between each and the other's trace-free subspaces and mutually …
Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension $n$, arbitrary polynomial degree $r$, and arbitrary differential form degree $k$. The study of finite element exterior calculus began with the $\mathcal P_rΛ^k$ and $\mathcal P_r^-Λ^k$ families of finite element spaces on simplicial triangulations. In their development of these spaces, Arnold, Falk, and Winther rely on a duality relationship between $\mathcal P_rΛ^k$ and $\mathri…
University of Cologne（科隆大学）；Universidad Politécnica de Madrid（马德里理工大学）；Max Planck Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）；Technical University of Munich（慕尼黑工业大学）；University of Waterloo（滑铁卢大学）
This article discusses a mixed FE technique for 3D nonlinear elasticity using a Hu-Washizu (HW) type variational principle. Here, the deformed configuration and sections from its cotangent bundle are taken as additional input arguments. The critical points of the HW functional enforce compatibility of these sections with the configuration, in addition to mechanical equilibrium and constitutive relations. The present FE approximation distinguishes a vector from a 1-from, a feature not commonly found in FE approximations. This point of view permits us to construct finite elements with vastly sup…
As traditional machine learning tools are increasingly applied to science and engineering applications, physics-informed methods have emerged as effective tools for endowing inferences with properties essential for physical realizability. While promising, these methods generally enforce physics weakly via penalization. To enforce physics strongly, we turn to the exterior calculus framework underpinning combinatorial Hodge theory and physics-compatible discretization of partial differential equations (PDEs). Historically, these two fields have remained largely distinct, as graphs are strictly t…
In this paper we propose a novel approach to discretize linear port-Hamiltonian systems while preserving the underlying structure. We present a finite element exterior calculus formulation that is able to mimetically represent conservation laws and cope with mixed open boundary conditions using a single computational mesh. The possibility of including open boundary conditions allows for modular composition of complex multi-physical systems whereas the exterior calculus formulation provides a coordinate-free treatment. Our approach relies on a dual-field representation of the physical system th…
The exterior derivative and the mean value equality in $\mathbb{R}^n$
Instituto de Ciências Matemáticas e de Computação (ICMC), Universidade de São Paulo (USP)（圣保罗大学数学与计算科学学院）；Instituto de Matemática, Estatística e Computação Científica (IMECC), Universidade Estadual de Campinas (UNICAMP)（坎皮纳斯州立大学数学、统计与科学计算研究所）
A four-field mixed variational principle is proposed for large deformation analysis of Kirchhoff rods with a $C^0$ mixed FE approximations. The core idea behind the approach is to introduce a one-parameter family of points (the centerline) and a separate one-parameter family of orthonormal frames (the Cartan moving frame) that are specified independently. The curvature and torsion of the curve are related to the relative rotation of neighboring frames. The relationship between the frame and the centerline is then enforced at the solution step using a Lagrange multiplier (which plays the role o…
