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SUMMARY:Towards Finite Element Tensor Calculus - Kaibo Hu (University of E
 dinburgh)
DTSTART:20240208T150000Z
DTEND:20240208T160000Z
UID:TALK210115@talks.cam.ac.uk
CONTACT:Nicolas Boulle
DESCRIPTION:Finite Element Exterior Calculus (FEEC) provides a cohomology 
 framework for structure-preserving discretisation of a large class of PDEs
 . There has been a relatively mature FEEC theory with de Rham complexes fo
 r problems involving differential forms (skew-symmetric tensors) and vecto
 r fields. A canonical discretisation exists\, which has a discrete topolog
 ical interpretation and can be generalized to other discrete structures\, 
 e.g.\, graph cohomology. \n\nIn recent years\, there has been significant 
 interest in extending FEEC to tensor-valued problems with applications in 
 continuum mechanics\, differential geometry and general relativity etc. In
  this talk\, we first review the de Rham sequences and their canonical dis
 cretisation with Whitney forms. Then we provide an overview of some effort
 s towards Finite Element Tensor Calculus (FETC). On the continuous level\,
  we characterise tensors and differential structures using the Bernstein-G
 elfand-Gelfand (BGG) machinery and incorporate analysis. On the discrete l
 evel in 2D and 3D\, we discuss analogies of the Whitney forms and establis
 h their cohomology. A special case is Christiansen’s finite element inte
 rpretation of Regge calculus\, a discrete geometric scheme for metric and 
 curvature. Moreover\, we present a correspondence between BGG sequences\, 
 continuum mechanics with microstructure and Riemann-Cartan geometry. These
  efforts are in the direction of establishing a tensor calculus on triangu
 lation and potentially on other discrete structures.
LOCATION:Centre for Mathematical Sciences\, MR14
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