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Speaker: Daniele DiPietro, University of Montpellier, France
Title: Polytopal approximations of the de Rham complex
Abstract: The de Rham complex for three-dimensional domains is the sequence of spaces H1, Hcurl, Hdiv, and L2 connected by the vector calculus operators grad, curl, div. The well-posedness of an important class of partial differential equations (PDEs) is linked to the cohomology properties of this complex, which characterize the kernel of each operator depending on the topology of the domain (how many connected components does it have? How many tunnels cross it? How many voids does it encapsulate?). Emulating these properties at the discrete level is crucial in order to obtain stable discretizations of such PDEs. A classical way of doing so consists in using trimmed (Raviart—Thomas—Nédélec) finite elements [1,2]. These finite elements are, however, restricted to conforming meshes with elements of simple shape. In this talk I will illustrate the main principles to derive approximations of the de Rham complex that support much more general meshes, including general polyhedral elements and non-matching interfaces [3]. A crucial step consists in switching to a fully discrete formulation, where both spaces and operators in the de Rham complex are replaced by discrete counterparts obtained mimicking integration by parts formulas. The resulting Discrete de Rham framework thus provides a new approach to design inherently stable schemes for PDEs whose well-posedness relates to the de Rham complex. Originally designed in vector calculus notation, this framework has been recently extended to the de Rham complex of differential forms [4]
[1] Raviart, P. A. and Thomas, J. M. (1977). A mixed finite element method for 2nd order elliptic problems. In Galligani, I. and Magenes, E., editors, Mathematical Aspects of the Finite Element Method. Springer, New York.
[2] Nédélec, J.-C. (1980). Mixed finite elements in R3. Numer. Math., 35(3):315–341.
[3] D. A. Di Pietro and J. Droniou. An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency. Found. Comput. Math., 2023, 23:85–164. DOI: 10.1007/s10208-021-09542-8
[4] F. Bonaldi, D. A. Di Pietro, J. Droniou, and K. Hu. An exterior calculus framework for polytopal methods. J. Eur. Math. Soc., 2024. Accepted for publication