[Dottorcomp] Seminari di Matematica Applicata, 26/05/2025, Matteo Ferrari, Kaibo Hu

[Stefano Lisini]

Seminari di Matematica Applicata, Dipartimento di Matematica "F. Casorati"
e Istituto del CNR IMATI "E. Magenes" di Pavia.

*Lunedì *26 Maggio 2025, alle *ore 14 precise*, presso la sala conferenze
dell'IMATI di Pavia,

Matteo Ferrari (Università di Vienna)

terrà un seminario dal titolo:Conforming space-time variational
formulations for the wave equation,

e alle *ore 15 precise*,

Kaibo Hu (University of Edinburgh)
terrà un seminario dal titolo:

Finite element form-valued forms.
----------------------------

Abstract (Ferrari): In this talk, we consider space-time conforming
Galerkin discretizations of the acoustic wave equation. Unlike
time-stepping methods, the time variable here is treated as an additional
dimension.
Our goal is to design numerical schemes that are unconditionally stable,
quasi-optimally convergent, and suitable for efficient implementation.
Ideally, we aim for a formulation that guarantees these properties under
minimal assumptions on the discrete spaces, allowing for broad
applicability. In particular, we are interested in methods that go beyond
piecewise continuous polynomials, extending also to spline functions of
arbitrary regularity.
We compare the properties, advantages, and limitations of three continuous
variational formulations, distinguished by their treatment of the temporal
part:
-a second-order-in-time scheme without integration by parts in time,
-a first-order-in-time scheme,
-a second-order-in-time scheme with integration by parts in time.
We will present recent results and highlight some open problems.
This talk is based on joint works with I. Perugia and E. Zampa.

-------------------

Abstract (Hu): Classical finite element methods, such as those by Lagrange,
Nédélec, Raviart–Thomas, and Brezzi-Douglas-Marini, fit within de Rham
complexes
and can be interpreted as discrete differential forms. These finite element
differential forms encode discrete topology and have become standard
practice for solving vector-valued problems. Their structures also find
broad applications in discrete topology, including topological data
analysis and the Hodge Laplacian on graphs.

In this work, we focus on tensors with applications in continuum mechanics,
differential geometry, and general relativity. First, weinvestigate the
algebraic and differential structures of tensor fields. We show that tensor
fields with natural symmetries fit within Bernstein-Gelfand-Gelfand (BGG)
complexes and twisted de Rham complexes, and we discuss the correspondence
between these complexes, generalized continua, and Riemann-Cartan geometry.
Second, we construct
finite elements for form-valued forms (double forms). Special cases include
classical finite element differential forms, distributional finite
elements, Christiansen’s finite element interpretation for Regge calculus
in quantum and numerical gravity (discrete metric and curvature), the
TDNNS/HHJ element for elasticity, the MCS element for Stokes equations, and
various new spaces.

References:

[1] Arnold, D. N., & Hu, K. (2021). Complexes from complexes. Foundations
of Computational Mathematics, 21(6), 1739-1774.

[2] Čap, A., & Hu, K. (2024). BGG sequences with weak regularity and
applications. Foundations of Computational Mathematics, 24(4), 1145-1184.

[3] Hu, K., Lin, T., & Zhang, Q. (2025). Distributional Hessian and divdiv
complexes on triangulation and cohomology. SIAM Journal on Applied Algebra
and Geometry, 9(1), 108-153.

[4] Hu, K., & Lin, T. (2025). Finite element form-valued forms (I):
Construction. arXiv preprint arXiv:2503.03243.

-----------------------------------------
Pagina web Seminari di Matematica Applicata
https://matematica.unipv.it/ricerca/cicli-di-seminari/seminari-di-matematica-applicata/