Related papers: Finite element exterior calculus: from Hodge theor…
Differential complexes such as the de Rham complex have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. The design of stable discretizations of systems of partial…
A low-order nonconforming finite element discretization of a smooth de Rham complex starting from the $H^2$ space in three dimensions is proposed, involving an $H^2$-nonconforming finite element space, a new tangentially continuous…
We want to propose a new discretization ansatz for the second order Hessian complex exploiting benefits of isogeometric analysis, namely the possibility of high-order convergence and smoothness of test functions. Although our approach is…
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…
This article surveys research on the application of compatible finite element methods to large scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa's C-grid finite difference scheme to the finite element…
We elaborate on the interpretation of some mixed finite element spaces in terms of differential forms. First we develop a framework in which we show how tools from algebraic topology can be applied to the study of their cohomological…
This article studies structure-preserving discretizations of Hilbert complexes with nonconforming spaces that rely on projections onto an underlying conforming subcomplex. This approach follows the conforming/nonconforming Galerkin (CONGA)…
We present a finite element scheme for fractional diffusion problems with varying diffusivity and fractional order. We consider a symmetric integral form of these nonlocal equations defined on general geometries and in arbitrary bounded…
Compatible discretizations, such as finite element exterior calculus, provide a discretization framework that respect the cohomological structure of the de Rham complex, which can be used to systematically construct stable mixed finite…
We discuss the construction of finite element spaces of differential forms which satisfy the crucial assumptions of the finite element exterior calculus, namely that they can be assembled into subcomplexes of the de Rham complex which admit…
This paper addresses the analysis and numerical assessment of a computational method for solving the Cahn--Hilliard equation defined on a surface. The proposed approach combines the stabilized trace finite element method for spatial…
We introduce a new finite element (FE) discretization framework applicable for covariant split equations. The introduction of additional differential forms (DF) that form pairs with the original ones permits the splitting of the equations…
Complexes of discrete distributional differential forms are introduced into finite element exterior calculus. Thus we generalize a notion of Braess and Sch\"oberl, originally studied for a posteriori error estimation. We construct…
We describe algorithms for finding harmonic cochains, an essential ingredient for solving elliptic partial differential equations in exterior calculus. Harmonic cochains are also useful in computational topology and computer graphics. We…
We present a finite element discretization of a non-linear diffusion equation used in the field of critical phenomena and, more recently, in the context of Dynamic Density Functional Theory. The discretized equation preserves the structure…
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…
This work develops a convergence theory for H(div)-conforming finite element methods applied to the steady Oseen problem, focusing on cases where the exact finite element complex holds while the commuting diagram property may fail. The…
We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously…
This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a…
We survey recent contributions to finite element exterior calculus on manifolds and surfaces within a comprehensive formalism for the error analysis of vector-valued partial differential equations on manifolds. Our primary focus is on…