50 research outputs found
Constructions of some minimal finite element systems
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 contain prescribed functions, and are minimal
under these constraints. We show how various known mixed finite element spaces
fulfill such a design principle, including trimmed polynomial differential
forms, serendipity elements and TNT elements. We also comment on virtual
element methods and provide a dimension formula for minimal compatible finite
element systems containing polynomials of a given degree on hypercubes.Comment: Various minor changes, based on suggestions of paper referee
Foundations of finite element methods for wave equations of Maxwell type
The first part of the paper is an overview of the theory of approximation of wave equations by Galerkin methods. It treats convergence theory for linear second order evolution equations and includes studies of consistency and eigenvalue approximation.
We emphasize differential operators, such as the curl, which have large kernels and use L2 stable interpolators preserving them.
The second part is devoted to a framework for the construction of finite element spaces of differential forms on cellular complexes.
Material on homological and tensor algebra as well as differential and discrete geometry is included. Whitney forms, their duals, their high order versions, their tensor products and their hp-versions all fit
Upwinding in finite element systems of differential forms
We provide a notion of finite element system, that enables the construction spaces of differential forms, which can be used for the numerical solution of variationally posed partial difeerential equations. Within this framework, we introduce a form of upwinding, with the aim of stabilizing methods for the purposes of computational uid dynamics, in the vanishing viscosity regime.
Published as the Smale Prize Lecture in: Foundations of computational mathematics, Budapest 2011, London Mathematical Society Lecture Note Series, 403, Cambridge University Press, 2013
On the linearization of Regge calculus
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 curlTcurl operator which appears in the
quadratic part of the Einstein-Hilbert action and also in the linear elasticity
complex. We insert Regge metrics in a discrete version of this complex,
equipped with densely defined and commuting interpolators. We show that the
eigenpairs of the curlTcurl operator, approximated using the quadratic part of
the Regge action on Regge metrics, converge to their continuous counterparts,
interpreting the computation as a non-conforming finite element method.Comment: 26 page
Stability of an upwind Petrov Galerkin discretization of convection diffusion equations
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. We identify norms for which we have both continuity and
an inf-sup condition, which are uniform in mesh-width and viscosity, up to a
logarithm, as long as the viscosity is smaller than the mesh-width or the
crosswind diffusion is smaller than the streamline diffusion. The analysis
allows for the formation of a boundary layer.Comment: v1: 18 pages. 2 figures. v2: 22 pages. Numerous details added and
completely rewritten final proof. 8 pages appendix with old proo
Extended Regge complex for linearized Riemann-Cartan geometry and cohomology
We show that the cohomology of the Regge complex in three dimensions is
isomorphic to , the
infinitesimal-rigid-body-motion-valued de~Rham cohomology. Based on an
observation that the twisted de~Rham complex extends the elasticity (Riemannian
deformation) complex to the linearized version of coframes, connection 1-forms,
curvature and Cartan's torsion, we construct a discrete version of linearized
Riemann-Cartan geometry on any triangulation and determine its cohomology.Comment: 24 page
A discrete elasticity complex on three-dimensional Alfeld splits
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 the linearized deformation operator, the linearized curvature operator, and the divergence operator, respectively. The construction is based on an algebraic machinery that derives the elasticity complex from de Rham complexes, and smoother finite element differential forms
A simplicial gauge theory
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 consistency. A discrete Noether's theorem that can be
applied to our setting, is also proved.Comment: 24 pages. v2: New version includes a longer introduction and a
discrete Noether's theorem. v3: Section 4 on Noether's theorem has been
expanded with Proposition 8, section 2 has been expanded with a paragraph on
standard LGT. v4: Thorough revision with new introduction and more background
materia