


Topology and Related Topics Seminar
Tuesday, 7 November 2023, 2:30 3:45 pm
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Venue: Ramanujan hall
Host: Rekha Santhanam
Speaker: Sayed Sadiqul Islam
Affiliation: IIT Bombay
Title: Kozul Complexes
Abstract: We'll begin by discussing the Koszul complex and regular rings. Then, we'll look into some fundamental properties of these ideas. Afterward, we'll explain the necessary and sufficient conditions that make a Noetherian local ring regular, without diving into complex proofs.
Analysis of PDE Seminar
Tuesday, 07 November 2023, 4:00 pm
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Venue: Room 113, Department of Mathematics
Host: Neela Nataraj
Speaker: Ricardo Ruiz Baier
Affiliation: Monash University
Title: New mixed finite element formulations for the coupled Stokes /PoissonNernstPlanck equations
Abstract: I will discuss a Banach spacesbased framework and new mixed finite element methods for the numerical solution of the coupled Stokes and PoissonNernstPlanck equations (a nonlinear model describing the dynamics of electrically charged incompressible fluids). The pseudostress tensor, the electric field (rescaled gradient of the potential) and total ionic fluxes are used as new mixed unknowns. The resulting fully mixed variational formulation consists of two saddlepoint problems, each one with nonlinear source terms depending on the remaining unknowns, and a perturbed saddlepoint problem with linear source terms, which is in turn
additionally perturbed by a bilinear form. The wellposedness of the continuous formulation is a consequence of a fixedpoint strategy in combination with the Banach theorem, the Babu\v{s}kaBrezzi theory, the solvability of abstract perturbed saddlepoint problems, and the BanachNe\v{c}asBabu\v{s}ka theorem. An analogous approach (but using now both the Brouwer and Banach theorems and stability conditions on arbitrary FE subspaces) is employed at the discrete level. A priori error estimates are derived, and examples of discrete spaces that fit the theory, include, e.g., RaviartThomas elements of order $k$ along with piecewise polynomials of degree $\le k$. Finally, several numerical experiments confirm the theoretical error bounds and illustrate the
balancepreserving properties and applicability of the proposed family of
methods.