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 /Poisson-Nernst-Planck equations

Abstract: I will discuss a Banach spaces-based framework and new mixed finite element methods for the numerical solution of the coupled Stokes and Poisson--Nernst--Planck 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 saddle-point problems, each one with nonlinear source terms depending on the remaining unknowns, and a perturbed saddle-point problem with linear source terms, which is in turn
additionally perturbed by a bilinear form. The well-posedness of the continuous formulation is a consequence of a fixed-point strategy in combination with the Banach theorem, the Babu\v{s}ka--Brezzi theory, the solvability of abstract perturbed saddle-point problems, and the Banach--Ne\v{c}as--Babu\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., Raviart--Thomas elements of order $k$ along with piecewise polynomials of degree $\le k$. Finally,  several numerical experiments confirm the theoretical error bounds and illustrate the
balance-preserving properties and applicability of the proposed family of
methods.