# Registered Data

## [00276] Interplay of Numerical and Analytical Methods in Nonlinear PDEs

**Session Date & Time**:- 00276 (1/2) : 1C (Aug.21, 13:20-15:00)
- 00276 (2/2) : 1D (Aug.21, 15:30-17:10)

**Type**: Proposal of Minisymposium**Abstract**: Devising reliable numerical schemes and analytically understanding fine properties of solutions of nonlinear partial differential equations are challenging mathematical tasks. Theoretically and practically relevant examples are geometrically constrained PDEs such as harmonic maps and isometric bending problems. Modern applications arise in the development of new storage technologies and micro tools. Numerical simulations provide valuable experimental insight that can motivate analytical results, e.g. about singularities. Conversely, stability results for solutions lead to convergence theories for numerical schemes. The minisymposium aims at bringing together scientists from analysis and numerics working on nonlinear PDEs in order to inspire new mathematical developments.**Organizer(s)**: Sören Bartels, Diane Guignard, Christof Melcher**Classification**:__35Bxx__,__65Nxx__,__49Jxx__**Speakers Info**:- Harbir Antil (George Mason University)
- Carlos Garcia-Cervera (UC Santa Barbara)
- Chunxi Jiao (UNSW Sydney)
- Alex Kaltenbach (University of Freiburg)
- Omar Lakkis (University of Sussex)
- Jiashun Hu (The Hong Kong Polytechnic University)
- Endre Süli (Oxford University)
- Shuo Yang (Tsinghua University)

**Talks in Minisymposium**:**[01330] Error analysis for a local discontinuous Galerkin approximation for systems of p-Navier–Stokes type****Author(s)**:**Alex Kaltenbach**(University of Freiburg)

**Abstract**: In this talk, we propose a Local Discontinuous Galerkin (LDG) approximation for systems of p-Navier–Stokes type involving a new numerical flux in the stabilization term and a new discretization of the convective term. A priori error estimates are derived for the velocity, which are optimal for all $p>2$ and $\delta\ge 0$. A new criterion is presented that yields a priori error estimates for the pressure, which are optimal for all $p>2$ and $\delta\ge 0$.

**[01583] Evolving FEMs with artificial tangential velocities for curvature flows****Author(s)**:**Jiashun Hu**(Hongkong Polytechnic University)- Buyang Li (Hongkong Polytechnic University)

**Abstract**: By considering a limiting situation in the method proposed by Barrett, Garcke and Nurnberg, a new artificial tangential velocity is introduced into the evolving finite element methods for mean curvature flow and Willmore flow to improve the mesh quality of the numerically computed surfaces. Stability and optimal-order convergence of the evolving finite element methods are established.

**[01636] Convergent finite element approximation of liquid crystal polymer networks****Author(s)**:**Shuo Yang**(Yanqi Lake Beijing Institute of Mathematical Sciences and Applications)- Ricardo Nochetto (University of Maryland)
- Lucas Bouck (University of Maryland)

**Abstract**: Liquid crystals polymer networks $(\text{LCN})$ deform spontaneously upon temperature or optical actuation. In this talk, we discuss a $2D$ membrane model of LCN and its properties. We design a finite element discretization for this model, propose a novel iterative scheme to solve the non-convex discrete minimization problem, and prove stability of the scheme and a convergence of discrete minimizers. We present a wide range of numerical simulations.

**[02913] Uniform flow in axisymmetric devices through permeability optimization****Author(s)**:**Harbir Antil**(George Mason University)- Drew P Kouri (Sandia National Labs)
- Denis Ridzal (Sandia National Labs)
- David Robinson (Sandia National Labs)
- Maher Salloum (Sandia National Labs)

**Abstract**: Porous media enable the intimate contact between a fluid and a functional solid that can accomplish tasks valuable to chemical engineers, such as catalytic reaction, chemical separations, chemical species detection, and filtration. New additive manufacturing technologies enable the creation of porous media with precise control of the geometry of each pore, which could enable improved performance and more flexible design of chemical engineering devices. However, new design tools are needed to accomplish this. In this talk, we analyze an optimization problem, constrained by Darcy's law, to design porous media columns that achieve uniform fluid flow properties despite having nonuniform geometries. We prove existence of solutions to our problem, as well as differentiability, which enables the use of rapidly converging, derivative-based optimization methods. We demonstrate our approach on two axisymmetric columns where we achieve a desired velocity field with uniform transit times despite varying device cross sections.

**[03481] Finite element approximation of implicitly constituted non-Newtonian fluids****Author(s)**:**Endre Suli**(University of Oxford)

**Abstract**: The framework of classical continuum mechanics, built upon an explicit constitutive equation for the stress tensor, is too narrow to describe inelastic behaviour of solid-like materials or viscoelastic properties of materials. We present a survey of recent results concerning the mathematical analysis of finite element approximations of implicit power-law-like models for viscous incompressible fluids, where the stress tensor and the symmetric part of the velocity gradient are related by a, possibly multi-valued, maximal monotone graph.

**[03482] A least squares Hessian/Gradient recovery method for fully nonlinear PDEs in Hamilton--Jacobi--Bellman form****Author(s)**:**Omar Lakkis**(University of Sussex)- Amireh Mousavi (Jena Universität)

**Abstract**: Least squares recovery methods provide a simple and practical way to approximate linear elliptic PDEs in nondivergence form where standard variational approach either fails or requires technically complex modifications. This idea allows the creation of relatively efficient solvers for fully nonlinear elliptic equations, the linearization of which leaves us with an equation in nondivergence form. An important class of fully nonlinear elliptic PDEs is that of Hamilton--Jacobi--Bellman form. Suitable functional spaces and penalties in the cost functional must be carefully crafted in order to ensure stability and convergence of the scheme with a good approximation of the gradient and Hessian which is useful, for example, for Newton--Raphson, semismooth Newton, or a policy iteration (Howard) approximation of a Hamilton--Jacobi--Bellman equation. We prove convergence and provide convergence rates under a Cordes condition.

**[03496] Regularised stochastic Landau-Lifshitz equations and their application in numerical analysis****Author(s)**:**Chunxi Jiao**(RWTH Aachen University )

**Abstract**: We revisit a regularised stochastic Landau-Lifshitz equation (sLLE) with bi-Laplacian in the effective field and study a similarly regularised stochastic Landau-Lifshitz-Bloch equation (sLLBE) in a two-dimensional domain. We derive the rate of convergence (in probability) of numerical solutions of a finite-element scheme for the regularised sLLBE to the solution of sLLBE, and outline the difficulty of applying this approach to sLLE. This talk is based on a joint work with Beniamin Goldys and Ngan Le.

**[05323] Hartree-Fock theory with a self-generated magnetic field****Author(s)**:**Carlos J. Garcia Cervera**(UCSB)- Rafael Lainez Reyes (UCSB)

**Abstract**: The study of a quantum system of N electrons interacting with K nuclei through the Coulomb potential has a long history in the mathematics community. In the first part of my talk, I will go over some of the quantum mechanical models developed to describe these systems, focusing on their mathematical structure and properties. Following that, I will describe how these theories change when a magnetic field is present. In particular, I will define the Hartree-Fock ground state problem for a system of N electrons and K nuclei in the presence of self-generated magnetic fields and direct coupling and we will study the existence of the ground state and excited states, as well as some numerical approaches for its computation. The work I present is in collaboration with Rafael Lainez Reyes.