# Numerical Analysis Seminar

## Fall 2023

Date: | September 20, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Johnny Guzman, Brown University |

Title: | Discrete elasticity complexes |

Abstract: | Mixed finite element methods for elasticity have several advantages including being robust in the nearly incompressible limit. Since the 2002 paper by Arnold and Winther there have been several inf-sup stable elasticity finite elements (a space for the stress and displacement) developed in two and three dimensions. However, finding an entire elasticity complex remained challenging in three dimensions. In the last few years, a few discrete elasticity complexes have been constructed. I will discuss our construction that uses macro-triangulations. This is joint work with Snorre Christiansen, Kaibo Hu, Sining Gong, Jay Gopalakrishnan, Michael Neilan. |

Date: | September 27, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Dionisios Margetis, University of Maryland |

Title: | On a non-Hermitian formalism for many-body Boson quantum dynamics |

Abstract: | The last three decades have witnessed interesting advances in atomic physics. Notably, the first experimental observation of a single macroscopic quantum state in trapped atomic gases, known as the phenomenon of ``Bose-Einstein condensation’’, at extremely low temperatures was reported in 1995. Since then, the efforts of physicists to harness cold atomic gases have expanded considerably. An emergent and far-reaching advance is the highly precise manipulation of atoms by optical or magnetic means in laboratory settings. In this talk, I will discuss mathematical implications of a physically motivated model for a dilute gas of zero-spin particles (Bosons) with repulsive pairwise interactions at zero temperature. In particular, I will describe aspects of the excited many-body quantum states of this system by accounting for the scattering of atoms in pairs from the macroscopic state (condensate). Key in this formulation is a non-unitary transformation of a prototypical many-body Hamiltonian. This transformation makes use of the ``pair-excitation kernel'', a function that satisfies a nonlinear partial integro-differential equation. For stationary tates, I will present an existence theory for solutions to this equation in a variational framework. I will also discuss how this theory is intimately connected to the physically motivated concept of ``quasiparticles’’, or collective excitations, in the atomic gas. This is joint work with M. Grillakis (UMD) and S. Sorokanich (NIST). |

Date: | October 18, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Sangyun Lee, Florida State University |

Title: | Physics Preserving Finite Element Methods for Coupled Multi-Physical Subsurface Applications |

Abstract: | In recent years, the main challenges in subsurface energy systems (e.g., enhanced geothermal systems and CO2 sequestration) have included issues arising from the multi-physical and multi-scale nature of the problem, as well as uncertainty quantification. Multi-physics involves coupling solid mechanics, fluid mechanics, thermal energy, and chemical reactions, while multi-scale considerations involve relating pore-scale problems to field-scale problems. These problems are complex and require interdisciplinary efforts to achieve meaningful outcomes in research. In this talk, we will focus on the challenges of the multi-physical formulations for solving subsurface applications, and discuss new enriched Galerkin (EG) finite element methods for coupled flow and transport and poro-elasticity systems. The primary goal of the study is to develop computationally efficient and robust numerical methods that are free from oscillations due to a lack of local conservation, maximum principle violations, int-sup issues, and locking effects. The locally conservative enriched Galerkin (LC-EG) method, which will be used to solve the flow problem, is constructed by adding a constant function to each element based on the classical continuous Galerkin methods (CG). The locking-free enriched Galerkin (LF-EG) method adds a piecewise linear vector to the displacement space. These EG methods incorporate well-known discontinuous Galerkin (DG) techniques but use approximation spaces with fewer degrees of freedom than typical DG methods, offering an efficient alternative. We will present a priori error estimates of optimal order and demonstrate, through numerical examples, that the new method is free from oscillations and locking. |

Date: | November 8, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Annalisa Quaini, University of Houston |

Title: | Towards the computational design of smart nanocarriers |

Abstract: | Membrane fusion is a potentially efficient strategy for the delivery of macromolecular therapeutics into the cell interior. However, existing nanocarriers formulated to induce membrane fusion suffer from a key limitation: the high concentrations of fusogenic lipids needed to cross cellular membrane barriers lead to toxicity in vivo. To overcome this limitation, we are developing in silico models that will explore the use of membrane phase separation to achieve efficient membrane fusion with minimal concentrations of fusion-inducing lipids and therefore reduced toxicity. The models we consider are formulated in terms of partial differential equations posed on evolving surfaces, i.e., the surface of the nanocarrier that undergoes fusion. For the numerical solution, we use a fully Eulerian hybrid (finite difference in time and trace finite element in space) discretization method. The method avoids any triangulation of the surface and uses a surface-independent background mesh to discretize the problem. Thus, our method is capable of handling problems posed on implicitly defined surfaces and surfaces undergoing strong deformations and topological transitions. |

Date: | November 15, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Yulong Xing, Ohio State University |

Title: | High Order Structure Preserving Numerical Methods for Euler Equations with Gravitation |

Abstract: | Hydrodynamical evolution in a gravitational field arises in many astrophysical and atmospheric problems. In this presentation, we will talk about high order structure preserving methods for the Euler equations under gravitational fields, which can exactly preserve some fundamental continuum properties of the underlying problems in the discrete level. We consider the Euler–Poisson equations in spherical symmetry with an equilibrium state governed by the Lane–Emden equation, and design well-balanced (WB) and total-energy-conserving (TEC) discontinuous Galerkin finite element methods. High order semi-implicit well-balanced asymptotic preserving (AP) finite difference scheme, for all Mach Euler equations with gravitation, may also be discussed. Extensive numerical examples — including a toy model of stellar core-collapse with a phenomenological equation of state that results in core-bounce and shock formation — are provided to verify the well-balanced property, positivity-preserving property, high-order accuracy, total energy conservation and good resolution for both smooth and discontinuous solutions. |

Date: | November 29, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Michael Neilan, University of Pittsburg |

Title: | Divergence-free finite element spaces for incompressible flow on isoparametric meshes |

Abstract: | In this talk, we construct and analyze an isoparametric and divergence-free finite element pair for the Stokes problem. The pair is defined by mapping the Scott-Vogelius finite element space via a non-traditional Piola transform. The velocity space has the same degrees of freedom as the canonical Lagrange finite element space, and the proposed spaces reduce to the Scott-Vogelius pair in the interior of the domain. The resulting method converges with optimal order, is divergence--free, and is pressure robust. In the second part of the talk, we extend this isoparametric framework to construct a stable and strongly conforming pair. This is achieved by constructing a new divergence-preserving (Piola) mapping combined with an enriching strategy that imposes full continuity across shared edges in the isoparametric mesh. |