报告地点：行健楼学术活动室526
邀请人：汪艳秋教授
报告摘要：Finite Element Exterior Calculus (FEEC) provides a cohomology framework for structure-preserving discretisation of a large class of PDEs. In FEEC, de Rham complexes for problems involving differential forms have been extensively discussed. A canonical construction of finite elements exists, which has a discrete topological interpretation and can be generalized to other discrete structures, e.g., graph cohomology.
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 discretisation with Whitney forms. Then we provide an overview of some efforts towards Finite Element Tensor Calculus. On the continuous level, we characterise algebraic and differential structures of tensors using the Bernstein-Gelfand-Gelfand (BGG) machinery. On the discrete level, we discuss analogies of the Whitney forms. A special case is Christiansen’s finite element interpretation of Regge calculus, a discrete geometric scheme for metric and curvature. We present a correspondence between algebra (BGG sequences), continuum mechanics (microstructure), Riemann-Cartan geometry (curvature and torsion) and discretization (de Rham’s currents).
