Abstract : Differential complexes encode important structures in a wide range of problems, and there has been a surge of interest in discretizing these complexes. Examples include the de-Rham complex, the elasticity complex, and, more recently, other BGG complexes. Algebraic and differential geometric structures play an important role in the construction of finite elements and multivariate splines. This mini-symposium aims to bring together researchers to discuss recent progress in the construction of discrete complexes and the emerging connections between algebra, geometry and discretization.
Daniele Antonio Di Pietro (University of Montpellier)
Jerome Droniou (Monash University)
Kaibo Hu (University of Oxford)
Abstract : For $\Omega\subset \mathbf{R}^n$, the de Rham complex of differential forms
$$
0\rightarrow H\Lambda^0(\Omega)\stackrel{d}{\rightarrow}H\Lambda^1(\Omega)\stackrel{d}{\rightarrow} \cdots
\stackrel{d}{\rightarrow} H\Lambda^n(\Omega)\stackrel{d}{\rightarrow}0
$$
is essential to establish the well-posedness of certain PDE models; developing discrete versions of this complex is a key for designing robust schemes for these models. We will present two such discrete complexes, inspired by the DDR and VEM approaches, of arbitrary order, and applicable on generic meshes. Compared to FEEC, these complexes benefit from the flexibility and high-level construction of polytopal methods.
[05349] Finite Element Complex
Format : Talk at Waseda University
Author(s) :
Long Chen (University of California at Irvine)
Xuehai Huang (Shanghai University of Finance and Economics)
Abstract : This presentation provides an overview of finite element complex construction, showcasing the finite element de Rham complex through a geometric decomposition method. The construction is extended to additional finite element complexes, such as the Hessian complex, elasticity complex, and divdiv complex, using the Bernstein-Gelfand-Gelfand (BGG) framework.
The resulting finite element complexes hold potential applications in numerical simulations for the biharmonic equation, linear elasticity, general relativity, and other geometry-related PDEs.
Abstract : Differential complexes have shed new insight into the finite
elements in recent years. This talk is devoted to the elasticity
complex, which provides an example of how complicated exact sequences
of spaces can be built from simple ones. Lining up two simpler
complexes, we start by performing a "diagram chase", which often goes
by the name of Bernstein-Gelfand-Gelfand resolution. The purpose of
this talk is to outline a few cases where this process can be
perfectly mimicked at the discrete level. The earliest example in
three-dimensions is on mesh of macroelements of Alfeld splits,
facilitated by the understanding of supersmoothness from research into
splines. Other emerging constructions will also be touched upon.
(Parts of the talk contain results obtained jointly with
S. Christiansen, S. Gong, J. Guzman, K. Hu, and M. Nielan.)
[03116] Nonconforming finite element exterior calculus
Format : Talk at Waseda University
Author(s) :
Shuo Zhang (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Abstract : A family of nonconforming finite element complexes are presented for n-dimensional de Rham complexes, $n\geq 2$. Particularly, the n-dimensional Crouzeix-Raviart elements are used to discretize the space of 0 forms. These finite element spaces are generally not constructed by Ciarlet's triples, and can be viewed as nonconforming splines. New theories are presented so that many basic properties of the finite element spaces and complexes can be established.
Abstract : For a planar simplicial complex Delta contained in R^2, Schumaker proved that a lower bound on the dimension of the space C^r_k(Delta) of planar splines of smoothness r and polynomial degree at most k on Delta is given by a polynomial P_Delta(r,k), and Alfeld-Schumaker showed this polynomial gives the correct dimension when k >= 4r+1. We prove that the equality dim C^r_k(Delta)= P_Delta(r,k) cannot hold in general for k <= (22r+7)/10.
[04982] Conforming Finite Element Methods with Arbitrary Smoothness in Any Dimension
Format : Talk at Waseda University
Author(s) :
Jun Hu (Peking University)
Ting Lin (Peking University)
Qingyu Wu (Peking University)
Abstract : This talk proposes a construction of $C^r$ conforming finite element spaces with arbitrary $r$ in any dimension. It is shown that if $k ≥ 2^d r + 1$ the space $P_k$ of polynomials of degree $≤ k$ can be taken as the shape function space of $C^r$ finite element spaces in $d$ dimensions. This is the first work on constructing such $C^r$ conforming finite elements in any dimension in a unified way.
[03133] An algebraic framework for geometrically continuous splines
Format : Talk at Waseda University
Author(s) :
Angelos Mantzaflaris (Inria at Universite Cote d'Azur, Sophia Antipolis, France)
Bernard Mourrain (Inria at Universite Cote d'Azur, Sophia Antipolis, France)
Nelly Villamizar (Swansea University)
Beihui Yuan (Swansea University)
Abstract : Geometrically continuous splines are piecewise polynomials defined on a collection of patches stitched together through transition maps. In this talk, we introduce an algebraic framework to study geometrically continuous splines. This framework enables us to use algebraic tools to analysis the dimension of spline spaces, and to present a new algorithm to construct bases using algebraic methods. This talk is based on a joint work with Angelos Mantzaflaris, Bernard Mourrain and Nelly Villamizar.
[05200] Multivariate spline functions on "oranges"
Format : Talk at Waseda University
Author(s) :
Nelly Villamizar (Swansea University)
Maritza Sirvent (The Ohio State University)
Tatyana Sorokina (Towson University)
Beihui Yuan (Swansea University)
Michael DiPasquale (University of South Alabama)
Abstract : A spline is a piecewise polynomial function defined on a partition of a real domain. Splines play an important role in many areas such as finite elements, computer-aided design, and data fitting.
In the talk, we will focus on splines defined on "oranges" which are partitions composed of a finite number of simplices of the same dimension that share one common lower dimensional face. For any fixed maximal polynomial degree and minimum order of global smoothness, we prove that the dimension of the spline space on an orange can be computed as a sum of the dimension of spline spaces on simpler lower-dimensional partitions. The examples and results in the atalk combine both Bernstein-Bézier methods for splines and algebraic tools.
[05202] The strain Hodge Laplacian and discretisation of the incompatibility operator
Format : Talk at Waseda University
Author(s) :
Francis Raul Anthony Aznaran (University of Oxford)
Kaibo Hu (University of Oxford)
Abstract : Motivated by the physical relevance of many Hodge Laplace PDEs from the FEEC, we analyse the Hodge Laplacian arising from the strain space $H(\mathrm{inc};\mathbb{R}^{d\times d}_{\mathrm{sym}})$, $\mathrm{inc} := \mathrm{rot}\circ\mathrm{rot}$, in the elasticity complex. We propose an adaptation of $C^0$-interior penalisation for the incompatibility, using the Regge element to discretise the strain. Building on pioneering work by van Goethem, we discuss promising connections between functional analysis of the $\mathrm{inc}$ operator and Kröner's intrinsic theory of defect elasticity.
[02874] Nonconforming finite elements for the Brinkman Problems and Quad-curl Problems on Cubical Meshes
Format : Talk at Waseda University
Author(s) :
Qian Zhang (Michigan Technological University)
Abstract : In this talk, I will present two families of nonconforming elements on cubical meshes: one for the quad-curl problem and the other for the Brinkman problem. The element for the quad-curl problem is the first nonconforming element on cubical meshes. The element for the Brinkman problem can yield a uniformly stable finite element method with respect to the parameter nu. The lowest-order elements for the quad-curl and the Brinkman problems have 48 and 30 degrees of freedom, respectively. The two families of elements, as a nonconforming approximation to H((gradcurl)) and H1, can form a discrete Stokes complex together with the Lagrange element and the DG element.
[05642] Distributional finite element BGG complexes
Format : Talk at Waseda University
Author(s) :
Jay Gopalakrishnan (Portland State University)
Kaibo Hu (University of Oxford)
Ting Lin (Peking University)
Joachim Schöberl (TU Wien)
Qian Zhang (Michigan Technological University)
Abstract : Distributional finite elements generalize classical concepts by allowing measures (Dirac deltas) as shape functions. Distributional elements were used to derive equilibrated residual error estimators (Braess, Schöberl 2008) and discretization of the stress-displacement formulation of linear elasticity (Pechstein, Schöberl 2011, TDNNS). Regge calculus from discrete relativity can be interpreted as a finite element version of metric with distributional curvature (Christiansen 2008). In this talk, we review the concept of distributional elements and discuss progress in discretizing Bernstein-Gelfand-Gelfand (BGG) diagrams and sequences.
