130 citations · 223 across the 20 of their papers we have counts for
21 papers · 1 filter
A note on short and long exact sequences in the BGG construction of complexes from complexes
Snorre H. Christiansen
We first show how the cohomology of some Bernstein-Gelfand-Gelfand (BGG) sequences that are important for the numerical analysis of partial differential equations, can be obtained…
Regge metrics with enhanced trace
Snorre H. Christiansen, Ting Lin
We introduce variants of Regge finite element metrics with enhanced properties of the trace. In particular the trace operator is surjective to a finite element space of continuous…
Interpretation of a Discrete de Rham method as a Finite Element System
Snorre H. Christiansen, Francesca Rapetti
We show that the DDR method can be interpreted as defining a computable consistent discrete product on a conforming FES defined by PDEs. Without modifying the numeri…
Convergence and Stability of Discrete Exterior Calculus for the Hodge Laplace Problem in Two Dimensions
Chengbin Zhu, Snorre H. Christiansen, Kaibo Hu +1
We prove convergence and stability of the discrete exterior calculus (DEC) solutions for the Hodge-Laplace problems in two dimensions for families of meshes that are non-degenerate…
Extended Regge complex for linearized Riemann-Cartan geometry and cohomology
Snorre H. Christiansen, Kaibo Hu, Ting Lin
We show that the cohomology of the Regge complex in three dimensions is isomorphic to , the infinitesimal-rigid…
A discrete elasticity complex on three-dimensional Alfeld splits
Snorre H. Christiansen, Jay Gopalakrishnan, Johnny Guzmán +1
We construct conforming finite element elasticity complexes on the Alfeld splits of tetrahedra. The complex consists of vector fields and symmetric tensor fields, interlinked via t…
Application of a minimal compatible element to incompressible and nearly incompressible continuum mechanics
Erik Burman, Snorre Christiansen, Peter Hansbo
In this note we will explore some applications of the recently constructed piecewise affine, -conforming element that fits in a discrete de Rham complex [Christiansen and Hu,…
Finite Element Systems for vector bundles : elasticity and curvature
Snorre H. Christiansen, Kaibo Hu
We develop a theory of Finite Element Systems, for the purpose of discretizing sections of vector bundles, in particular those arizing in the theory of elasticity. In the presence…
Nodal Finite Element de Rham Complexes
Snorre H. Christiansen, Jun Hu, Kaibo Hu
We construct 2D and 3D finite element de Rham sequences of arbitrary polynomial degrees with extra smoothness. Some of these elements have nodal degrees of freedom (DoFs) and can b…
Generalized Finite Element Systems for smooth differential forms and Stokes problem
Snorre Harald Christiansen, Kaibo Hu
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 framewor…
On eigenmode approximation for Dirac equations: differential forms and fractional Sobolev spaces
Snorre H. Christiansen
We comment on the discretization of the Dirac equation using finite element spaces of differential forms. In order to treat perturbations by low order terms, such as those arizing…
Second order gauge invariant discretizations to the Schrödinger and Pauli equations
Snorre Harald Christiansen, Tore Gunnar Halvorsen
We introduce a numerical method, based on finite elements and lattice gauge theory, to compute approximate solutions to Schrödinger and Pauli equations. The crucial geometric prope…
Constructions of some minimal finite element systems
Snorre H. Christiansen, Andrew Gillette
Within the framework of finite element systems, we show how spaces of differential forms may be constructed, in such a way that they are equipped with commuting interpolators and c…
Stability of an upwind Petrov Galerkin discretization of convection diffusion equations
Snorre H. Christiansen, Tore G. Halvorsen, Torquil M. Sørensen
We study a numerical method for convection diffusion equations, in the regime of small viscosity. It can be described as an exponentially fitted conforming Petrov-Galerkin method.…
Exact formulas for the approximation of connections and curvature
Snorre Harald Christiansen
First we express the holonomy along a boundary curve as the integral on the domain, of an expression which is linear in the curvature. Then we provide a rigorous justification of t…
On high order finite element spaces of differential forms
Snorre Harald Christiansen, Francesca Rapetti
We show how the high order finite element spaces of differential forms due to Raviart-Thomas-Nédelec-Hiptmair fit into the framework of finite element systems, in an elaboration of…
On the linearization of Regge calculus
Snorre H. Christiansen
We study the linearization of three dimensional Regge calculus around Euclidean metric. We provide an explicit formula for the corresponding quadratic form and relate it to the cur…
Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension
Snorre Harald Christiansen
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 topolo…
Finite element systems of differential forms
Snorre Harald Christiansen
We develop the theory of mixed finite elements in terms of special inverse systems of complexes of differential forms, defined over cellular complexes. Inclusion of cells correspon…
A simplicial gauge theory
Snorre Harald Christiansen, Tore Gunnar Halvorsen
We provide an action for gauge theories discretized on simplicial meshes, inspired by finite element methods. The action is discretely gauge invariant and we give a proof of consis…
On variational eigenvalue approximation of semidefinite operators
Snorre Harald Christiansen, Ragnar Winther
Eigenvalue problems for semidefinite operators with infinite dimensional kernels appear for instance in electromagnetics. Variational discretizations with edge elements have long b…