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Texas A&M University

Events for 11/09/2022 from all calendars

Noncommutative Geometry Seminar

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Time: 2:00PM - 3:00PM

Location: BLOC 302

Speaker: Dan Lee, Queens College CUNY

Title: The equality case of the spacetime positive mass theorem

Abstract: The spacetime positive mass theorem states that asymptotically flat initial data sets satisfying the dominant energy condition (a physical condition expressing nonnegativity of matter sources) must have “nonnegative mass” in the sense that the ADM energy-momentum vector (E,P) must be “future causal,” that is, E \ge |P|. This result goes back to Witten in the spin case and Schoen-Yau and Eichmair-Huang-Lee-Schoen for manifolds with dimension less than 8. It was always conjectured that the equality E=|P| should only be possible for initial data sets arising from slices of Minkowski space, but it is surprisingly tricky to prove. A rigorous proof in the spin case was not discovered until 15 years after Witten’s proof, by Beig-Chrusciel (n=3) and Chrusciel-Maerten (n>3). Recently, in joint work with Lan-Hsuan Huang, we built on some insights of Beig-Chrusciel to find a proof that depends only upon knowing that the inequality E \ge |P| holds for all nearby initial data sets that also satisfy the hypotheses of the spacetime positive mass theorem. Or in other words, our proof characterizing the equality case does not depend on *how* one proves the inequality.

Numerical Analysis Seminar

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Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Vladimir Yushutin, Clemson University

Title: T-Rex FEM: an abstract analysis framework for unfitted methods

Abstract: Unfitted, non-conforming finite element methods have the following in common: there is a drastic difference between the space of solutions and the finite element space. This difference manifests on the discrete level where one needs to employ a discrete stabilization form to guarantee the well-posedness of linear problems. Convergence analysis for such methods often follows the second Strang lemma, conditions of which may be hard to verify in some situations.

Instead, we study the strong convergence of unfitted continuous-in-time approximations via compactness. With this goal in mind, we develop an analysis framework, called T-Rex FEM, that involves notions of abstract TRace and EXtension operators. We build this analysis framework sequentially starting from abstract linear elliptic, parabolic, saddle problems and applying it to Navier--Stokes and Allen--Cahn equations.

The key ingredient is a problem-dependent modification of the abstract discrete stabilization form that makes the scheme amenable to a proof by compactness. We test numerically the modified scheme suggested by the T-Rex FEM when it is applied to the surface heat equation being solved by the Trace FEM - an unfitted method for surface PDEs which uses a bulk mesh surrounding the surface. In addition to the advantage of T-Rex FEM from the analysis standpoint, the new scheme restores the conditioning of linear problems, known in the fitted case for the heat equation, despite the presence of the stabilization form.