| Jan 19 | Jan 20 | Jan 21 | Jan 22 | Jan 23 | |
|---|---|---|---|---|---|
| Chairman | Ta-tsien Li | Weinan E | Yann Brenier | Hisashi Okamoto | Xiaoming Wang |
| Session 1 | Takis Souganidis | Hiroshi Matano | Shuxing Chen | Richard McLaughlin | Thomas Laurent |
Tea Break |
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| Chairman | Ta-tsien Li | Weinan E | Yann Brenier | Hisashi Okamoto | Xiaoming Wang |
| Session 2 | Weinan E | Marcus Grote | Shi Jin | Bjorn Engquist | Edriss Titi |
| Ilya Timofeyev | Zhouping Xin | Ping Zhang | Yann Brenier | Shangyou Zhang | |
Lunch |
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| Chairman | Zhouping Xin | Jiaxing Hong | Shi Jin | Sightseeing | Guiqiang Chen |
| Session 3 | Jialin Lin | Fang-Hua Lin | David Cai | Dongho Chae | |
| Joseph Biello | Peter Kramer | Yuxi Zheng | Takaaki Nishida | ||
| Diego Cordoba | |||||
| Tea Break |
Tea Break | ||||
| Chairman | Zhouping Xin | visiting Yuyuan Garden & Banquet | Shi Jin | Guiqiang Chen | |
| Session 4 | Jianbo Gao | Bruce Turkington | Xiaoming Wang | ||
| Rafail Abramov | Robert Pego | Denis Serre | |||
| Hisashi Okamoto | Markos Katsoulakis | Boualem Khouider | |||
Every speaker has 45 minutes for the speach, and there will be a tea break between 2 sessions.
Venue: Room 1801, East GuangHua Main Building
For detailed schedule, please check your conference booklet(pdf version) or just click here(webpage).
Last Update: 14:30, 2009-1-14
Title: Climate Response through Fluctuation-Dissipation: A New Algorithm for Low Frequency Dynamics
Abstract:
Recently, we developed and tested a novel computational algorithm for predicting the mean response of a chaotic dynamical system to small changes in external forcing via the fluctuation-dissipation theorem (FDT). Unlike the earlier work in developing FDT-based computational strategies for chaotic nonlinear systems with forcing and dissipation, the new method is based on the theory of SRB probability measures, which commonly describe equilibrium states of such dynamical systems. The new response algorithm is tested on a model of the barotropic climate with realistic Earth orography and forcing mimicking the dynamics of the atmosphere at 300 and 500 hPa geopotential height. The new method yields greater accuracy than the classical FDT method for the response of both mean state and variance for large scale EOFs. These results point the way toward the potential use of the new response algorithm in operational long-term climate change prediction.
Title: PDEs and Asymptotics for the Tropical Atmosphere
Abstract:
I shall discuss two interesting PDE problems for the tropical atmosphere which I have worked on with Andy Majda.
Using systematic multiscale asymptotics, Majda and Klein arrived at an asymptotic closure for the ideal fluid equations governing dynamics on large scales in the tropical atmosphere. Along with Majda, we considered a plausible model for smaller scale flows in the tropics and are able to calculate the structure of the Madden-Julian oscillation; this is a planetary scale organization of winds,the understanding of which has been called "the holy grail" of tropical meteorology.
In the second problem, we again study the equatorial primitive equations, but over longer time and spatial scales. The resultant coupled nonlinear dispersive equations for the amplitudes of interacting wave packets are novel both from the perspective of the atmospheric sciences and from a more general mathematical setting. These equations describe the influence of large scale tropical waves on midlatitude waves and, in particular, are relevant for understanding the effect of the Madden-Julian oscillation on midlatitude weather. We also discuss the Hamiltonian structure of these waves and show that they admit some analytic solitary wave solutions.
Title: Rearrangement and Convection: A Link between Analysis and Mechanics
Abstract:
Rearrangement theory is about reorganizing a given function (or map) in some specific order (monotonicity, cycle monotonicity etc...). This is vaguely reminiscent of the convection mechanism that rearranges fluid parcels in order to organize their position in a stabler way (heavy fluid at bottom and light fluid at top). In our talk, we make this analogy more precise by examining the relationship between (a generalized version of) the Navier-Stokes-Boussinesq system (that describes convection mechanisms) and the rearrangement of maps as gradient of convex functions. (Related topics include the Angenent-Haker-Tannenbaum model for optimal transportation and the Cullen-Purser stability criterion for Hoskins' semigeostrophic equations.)
Title: Renormalized Resonances and Wave Turbulence
Abstract:
We discuss a non-perturbative nature of wave turbulence and show how the wave spectrum n(k) of nonlinear dispersive waves is determined by an intertwining self-consistent process: The trivial resonant scatterings of waves off of background waves characterized by n(k) control the true, renormalized, dispersion relation. The renormalized dispersion relation, in turn, controls nontrivial resonances of the full wave system, thus, giving rise to a self-consistent wave spectrum n(k).
Title: Some Results on the Incompressible Euler Equations
Abstract:
In this talk I would like to present some of my results on the incompressible Euler equations in $R^3$. We begin with briefly reviewing the two classical results on the blow-up criterion due to Beale-Kato-Majda, and Constantin-Fefferman-Majda. Then, we show that these two criteria can be `unified' by introducing Triebel-Lizorkin space for the direction fields of vorticity. We also discuss nonexistence results of self-similar and asymptotically self-similar blow-up, and their recent refinements. Finally I would like to present Liouvile type results for the equ ons, which claim that suitable integrability condition of velocity together with sign condition of integrals of pressure implies vanishing of the velocity.
Title: to be announced
Abstract:
to be announced
Title: The Rayleigh-Taylor Condition for the Evolution of Irrotational Fluid Interfaces
Abstract:
We consider the free boundary dynamics between two irrotational, incompressible and inviscid fluids in 2-D without surface tension. For the two-phase Hele-Shaw and Muskat problems, and also for irrotational incompressible Euler equation in vacuum, we present a proof of the existence locally in time when the Rayleigh-Taylor condition is initially satisfied on the interface.
Title: Localized Basis of Eigen-Subspaces and Operator Compression
Abstract:
Given a complex local operator, such as the generator of a Markov chain on a large network, a differential operator or a large sparse matrix that comes from the discretization of a differential operator, we would like to find its best finite dimensional approximation with a given dimension. The answer to this question is often given simply by the projection of the original operator to its eigen-subspace of the given dimension that corresponds to the smallest or largest eigenvalues, depending on the setting. The representation of such subspaces, however is far from being unique and our interest is to find the most localized basis for these subspaces. The reduced operator using these bases would have sparsity features similar to that of the original operator. We will discuss different ways of obtaining localized basis, and we will give an explicit characterization of the decay properties of these basis functions. We will also discuss efficient numerical algorithms for finding such basis functions and the reduced (or compressed) operator, and applications to electronic structure analysis.
Title: Multiscale Simulations of Dynamical Systems with Oscillatory Solutions
Abstract:
The heterogeneous multiscale method is a framework that can be used to design and analyze numerical methods for stiff ordinary differential equations with oscillatory solutions. We will discuss the difficulty of finding macro-scale variables and also compare the heterogeneous multiscale method with related techniques. Applications to molecular dynamics will be considered.
Title: Quantifying Information Loss in Ensemble Forecasting of Dynamical Systems
Abstract:
Ensemble foresting is a widely used technique in numerical weather and climate predictions, to which Prof. Majda has been making significant and lasting contributions. After reviewing the status of the field, and especially Prof. Majda's work, I'll briefly present our own preliminary (and quite minor) contributions to the topic, 1) a pseudo-ensemble technique, which in certain situations may improve accuracy and expedite computations in ensemble forecasting of large dynamical systems, and 2) closed-form formulas, which can be used to quantify information loss in various types of dynamical systems.
Title: Nonreflecting Boundary Conditions for Time Dependent Multiple Scattering Problems
Abstract:
The accurate and reliable simulation of wave phenomena is of fundamental importance in a wide range of engineering applications from acoustics, electromagnetics and elasticity. For problems set in an unbounded domain, an artificial boundary is required to confine the region of interest to a finite computational domain. Then, a nonreflecting boundary condition is required at the artificial boundary, which avoids spurious reflections from it.
In 1977 Bjorn Engquist and Andrew Majda proposed the first hierarchy of local absorbing boundary conditions which approximate the necessarily nonlocal exact condition. Starting from their pioneering contribution we briefly review further developments in that rapidly growing research area, which in the late nineties led to convergent numerical methods for time dependent scattering problems.
Next, when a scatterer consists of several components, the use of a single artificial boundary to enclose the entire region of interest becomes too expensive. Instead, it is preferable to embed each component of the scatterer in a separate sub-domain. As waves bounce back and forth between domains, they are no longer purely outgoing outside the computational domain, so that most standard approaches cannot be used directly. We show how to overcome this difficulty and derive local nonreflecting boundary conditions for time dependent multiple scattering problems.
Title: An Eulerian Gaussian Beam Method for High Frequency Waves
Abstract:
The solution to high frequency waves (acoustic waves, elastic waves, quantum mechanics, electromagnetic waves, etc.) is computationally challenging due to the small wave length \epsilon. A direct numerical simulation requires the mesh size (and time step) to be O(\epsilon). The Gaussian beam method is an efficient way to solve the high frequency wave equations asymptotically, outperforming geometric optics method in that the Gaussian beam method is accurate even at caustics.
We describe our Gaussian beam method using the example of the Schrodinger equation in the semiclassical regime, where the scaled Planck constant epsilon is extremely small. Our new Eulerian Gaussian beam method is developed using the level set method based only on solving the (complex-valued) homogeneous Liouville equations. A major contribution here is that we are able to construct the Hessian matrices of the beams by using the level-set function's first derivatives. This greatly reduces the computational cost in computing the Hessian of the phase function in the Eulerian framework, yielding an Eulerian Gaussian beam method with computational complexity comparable to that of the geometric optics but with a
much better accuracy around caustics.
This is a joint work with Xu Yang and Hao Wu.
Title: Hierarchical and Multi-Level Coarse-Graining Methods
Abstract:
We will discuss a variety of coarse-graining methods for many-body microscopic systems. We focus on mathematical, numerical and statistical methods allowing us to assess the parameter regimes where such approximations are valid. We also demonstrate, with direct comparisons between microscopic (DNS) and coarse-grained simulations, that the derived mesoscopic models can provide a substantial CPU reduction in the computational effort. Furthermore, we discuss the feasibility of spatiotemporal adaptivity methods for the coarse-graining of microscopic simulations, having the capacity of automatically adjusting during the simulation if substantial deviations are detected in a suitable error indicator. Here we will show that in some cases the adaptivity criterion can be based on a posteriori estimates on the loss of information in the transition from a microscopic to a coarse-grained system. Finally, motivated by related problems in the simulation of macromolecular systems, we discuss mathematical strategies for reversing the coarse-graining procedure. The principal purpose of such a task is recovering local microscopic information in a large system by first employing inexpensive coarse-grained solvers.
Title: Simple Models for the Tropics: From the Parametrization of Convection to Some Saddle Numerical Issues
Abstract:
In this talk I will discuss some idealized models for the tropical atmosphere circulation, on which prof. Andrew Majda and I have been working for a few years now, to address some practical and theoretical issues in this field. The talk is divided into two parts. The first part is devoted to multicloud model parametrization for organized tropical convection. We use physical intuition as well as quantitative climate variables to design a simple convective parametrization for moist convection, which takes into account some key features of organized tropical convection. Consistent with the intuition, linear stability analysis reveal unstable modes reminiscent of convectively coupled waves shown by satellite observations. The second part of the talk addresses some saddle issues regarding nonlinear wave interactions in the tropics. We are especially interested in the interactions between vertical modes of different vertical structure and moisture. We emphasis on the design of robust numerical schemes for simple models obtained by vertical Galerkin projections of the nonlinear hydrostatic rotating Boussinesq equations, on an equatorial beta-plane that can capture some of the balanced properties of these waves with the minimum ad hoc dissipation. Among other interesting mathematical and physical features, some of the simple systems can actually lose hyperbolicity in some regions of the phase-plane, perhaps due to the ill-posedness of the primitive equations. Thus designing both robust and accurate numerical schemes for such systems can be challenging.
Title: Two Coarse-Graining Studies of Stochastic Models in Molecular Biology
Abstract:
We examine stochastic coarse-graining strategies for two biomolecular systems. First, we compute the large-scale transport properties of the basic flashing ratchet mathematical model for (Brownian) molecular motors and consider in this light whether the underlying continuous-space, continuous-time Markovian model can be coarse-grained as a discrete-state, continuous-time Markovian random walk model. Through careful computation of associated statistical signatures of Markovianity, we find that such a discrete coarse-graining is an excellent approximation over much but not all of the parameter regime. In particular, for the parameter values associated with the most efficient operation of the flashing ratchet, the discretized model displays non-Markovian features such as waiting times between jumps which are not exponentially distributed. We provide a theoretical framework for understanding the conditions under which Markovianity is to be expected in the discretized model and two mechanisms by which the flashing ratchet model coarse-grains to a non-Markovian discretized model. Next we turn to a basic question of how the dynamics of water molecules near the surface of a solute can be represented by a simple drift-diffusion stochastic model. This question is of most interest for the purpose of accelerating molecular dynamics simulations of proteins, but for simplicity, we here examine the simple case where the solute is a $ C_{60} $ buckyball, which has a homogenous, roughly isotropic form. We compare the mathematical drift-diffusion framework with a statistical quantification of water dynamics near a solute discussed in the biophysical literature. A key concern is the choice of time interval on which to sample the molecular dynamics data to generate estimators for the drift and diffusivity. We use a simple mathematical toy model to establish insight and a strategy, but find for the actual molecular dynamics data, that the sampling times which produce the most faithful drift coefficient and the sampling times which produce the most faithful diffusion coefficient do not overlap, so that sacrifice of quality in one or the other parameter appears necessary.
Title: Blowup in Multidimentional Aggregation Equation with Mildly Singular Interaction Potential.
Abstract:
We consider the conservation law u_t+div(uv)=0 where the velocity field is defined via a convolution: v=-grad K*u. The radially symmetric interaction potential K has a mild singularity at the origin (Lipschitz or better). We will determine the exact amount of singularity needed on the interaction potential in order to generate finite time blow-up. To be more precise, we will show that if K satisfies some Osgood condition, then solutions of the PDE exist for all time, whereas if the Osgood condition is not satisfied, the L^p norm of compactly supported solutions blow-up in finite time.
Abstract:
Title: Convectively Coupled Equatorial Waves Simulated by Global Climate Models
Abstract:
A fundamental contribution Prof. Majda has made to the atmospheric sciences is his pioneering theoretical study of the convectively coupled equatorial waves, the key modes affecting both short-term weather prediction and long-term climate prediction. This talk will provide a review of the convectively coupled equatorial waves simulated by the state-of-the-art global climate models, especially those participating in the Inter-governmental Panel on Climate Change (IPCC) Fourth Assessment Report (AR4). The modes we have examined include the Madden-Julian Oscillation (MJO), boreal summer intraseasonal oscillation (BSIO), Kelvin wave, equatorial Rossby (ER) wave, mixed Rossby-gravity (MRG) wave, eastward inertio-gravity (EIG) wave, westward inertio-gravity (WIG) wave, and tropical easterly waves. We will discuss about the common problems, the associated physical mechanisms, and the insights from Prof. Majda's theoretical studies.
Title: Finite Time Mixing and Entrainment in Flows with Fixed and Moving Boundarie
Abstract:
The fluid dynamics occurring in variable density, stratified systems is complicated by buoyancy generation and associated instabilities. Buoyancy quantification often requires prediction of fluid entrainment as well as its mixing. This is perhaps no simpler demonstrated than through the observation of a stone falling through a stably stratified pond. In this lecture, we'll examine experiments concerning the gravitational descent of spheres through a sharp stable stratification. We then focus our theoretical discussion on two related idealized problems with long histories, both originating in 1953. First, we re-examine the theory and experiments of G. I. Taylor who showed that a passive scalar in laminar pipe flow in a mean speed Galilean reference frame would experience an enhanced longitudinal mixing beyond the molecular diffusion in an amount U^2 a^2/(192 D) where U is the maximal flow, a is the pipe radius, and D is the dye's molecular diffusivity. This boosted diffusivity was predicted to arise after a transient timescale, a^2/D, the diffusion time to cross the pipe. We compute the exact evolution of the scalar variance yielding the short time anomalous behavior and the long time predictions of Taylor's multi-scale calculation. Second, we re-examine the predictions of Darwin who, assuming potential flow, connected the fluid volume entrained by a body moving a doubly infinite distance with the hydrodynamic added mass of the body. We focus on the case of a sphere, and compute finite time asymptotic corrections to the entrained fluid volume and along the way compute the asymptotic properties of a new fluid Brachistochrone path. We close with discussion of the potential energy associated with this entrained fluid, which provides a coarse arrestment criteria for a sphere falling through a stratified fluid. The first is joint with R. Camassa and Z. Lin, the second is joint with Camassa, M. Moore, and A. Vaidya.
Title: Front Propagation in Spatially Ergodic Media
Abstract:
A recurrent function is called ergodic if it has a certain averaging property. This concept is wider than and is often more natural than almost periodicity when one studies front propagation under spatially heterogeneous environtments.
In this talk, I will discuss the following subjects:
(1) speed of travelling waves in spatially ergodic media;
(2) long-time behavior of ergodically disturbed planar fronts in the Allen-Cahn equation;
(3) spreading fronts in spatially non-periodic diffusive media.
Title: Pattern Formations in Heat Convection Problems
Abstract:
Heat convection problems are considered in the horizontal strip domain where viscous fluids are heated from below under the gravity. The Oberbeck-Boussinesq equation is assumed to govern the fluid motion. The boundary condition on the upper surface is stress free and that on the lower boundary is no-slip. The bifurcation theorem gives some pattern formations. Beyond the bifurcation theorem we want to extend the bifurcated solution curves.
The case of free surface for the upper boundary ( Benard-Marangoni problem ) is also analyzed by a generalization of bifurcation theory.
Title: Stationary Vortices of Large Scale at Large Reynolds Numbers
Abstract:
We consider the sinusoidally forced Proudman-Johnson equation. Stability of and bifurcation from the trivial solution are studied numerically and analytically. We then compute solutions with large Reynolds numbers. With external forces of single, double, and triple mode, there exists always a family of stable solutions, which converges to a constant multiple of $\sin x$. Namely, despite the increase of the frequency of the external force, we can find a unimodal solution when the Reynolds number is very large. Whether this can be seen universally or not is difficult to see, but we verified this also in the Kolmogorov flows in 2D flat tori. This phenomena may well be connected with the inverse cascade theory of 2D turbulence.
Title: How to Compute the Pressure for Viscous Incompressible Flow with No-Slip Boundary
Abstract:
For the Navier-Stokes equations for incompressible flow with no-slip boundary conditions, we describe a formula for the pressure that involves the commutator of the Laplacian and Leray-Helmholtz projection operators. An estimate for this commutator shows that it is strictly dominated by the viscous term at leading order. This leads to a number of developments, including a simple well-posedness theorem for an extended Navier-Stokes dynamics unconstrained by the divergence-free condition, and improvements of methods of numerical computation and numerical analysis for such flows.
Title: Shock Reflection in Gas Dynamics
Abstract:
J. von Neumann and A. Majda popularized the analysis of multidimensional shock reflection in gas dynamics. A major issue is the reflection of a planar shock against either a planar wall or a solid wedge. We shall present qualitative and as well quantitative results on these problems. We give a definitive answer to the triple point question. We show that minimum/maximum principles yield pointwise estimates for the internal variables, the velocity and the size of the subsonic bubble in the Regular Reflection. We show however that a vortical singularity must exist.
Title: Molecular Motors and Asymmetric Potentials
Abstract:
I describe some mathematical models explaining uni-directional motions of molecular motors in the presence of asymmetric potentials.
Title: Mode Reduction Strategy for Systems with Scale Separation
Abstract:
In many areas of science and engineering it is often desirable to derive a closed-form reduced system describing the behavior of large-scale structures in chaotic high-dimensional systems. I will describe the development of the stochastic mode-reduction strategy for systems with scale separation. This technique allows reducing the dimensionality of the system and deriving a closed-form stochastic differential equation for the slow variables alone. Simple examples will be presented to illustrate this technique and discuss regimes of applicability. In addition, implications for the atmospheric fluid dynamics will also be discussed.
Joint Work : A. Majda, I. Timofeyev, E. Vanden-Eijnden
Title: On the Question of Global Regularity for Three-dimensional Navier-Stokes Equations and Relevant Geophysical Models
Abstract:
The basic problem faced in geophysical fluid dynamics is that a mathematical description based only on fundamental physical principles, the so-called the “Primitive Equations”, is often prohibitively expensive computationally, and hard to study analytically. In this talk I will survey the main obstacles in proving the global regularity for the three-dimensional Navier-Stokes equations and their geophysical counterparts. Even though the Primitive Equations look as if they are more difficult to study analytically than the three-dimensional Navier-Stokes equations I will show in this talk that they have a unique global (in time) regular solution for all initial data.
Inspired by this work I will also provide a new global regularity criterion for the three-dimensional Navier-Stokes equations involving the pressure.
This is a joint work with Chongsheng Cao.
Title: An Optimization Approach to Statistical Closure of Underresolved Hamiltonian Dynamics
Abstract:
A fundamental problem in applied mathematics is the derivation of closed reduced equations for complex, high-dimensional dynamical systems. Typically one wants to approximate the evolution of a few relevant variables rather than simulate all the variables that enter into the governing dynamics. Traditionally this problem is handled by some kind of projection technique, but often the resulting reduced equations involve expressions that are difficult or expensive to evaluate. I will describe a new approach to this general problem in the case of deterministic Hamiltonian dynamics. The key idea is to seek the "best-fit" statistical distribution to the Liouville equation from a convenient class of trial distributions. The fitting uses an information-theoretic metric integrated over the time horizon of the predicted evolution. The desired closure is accomplished by solving an optimization problem in the few variables that parameterize the trial distributions. The resulting reduced dynamics has the form of a nonequilibrium thermodynamics in the relevant macrovariables. In the near-equilibrium regime, the closure is defined by a time-dependent transport matrix which is derived completely from the underlying microscopic dynamics. After explaining the theory, I will show some numerical tests of the predictive macroscopic equations.
Title: Approximating Stationary Statistical Properties in Dissipative Chaotic Dynamical Systems
Abstract:
It is well-known that physical laws for large chaotic systems are revealed statistically. We consider temporal approximation of stationary statistical properties of dissipative chaotic dynamical systems. We demonstrate that appropriate time discretization viewed as discrete dynamical system is able to capture asymptotically the stationary statistical properties of the underlying continuous dynamical system provided that two conditions are satisfied: (1) The discrete dynamical system inherits the dissipativity of the original system uniformly (with respect to time step) in some appropriate sense; (2) The discrete dynamical system converges uniformly on the unit time interval [0,1] to the original system uniformly for initial data coming from the union of the global attractors. Application to the infinite Prandtl number model for convection will be discussed.
Title: Mixed-Typed Equations and Compressible Flows in Nozzles
Abstract:
In this talk, I will survey some of recent results on ideal steady compressible flows in general nozzles with variable sections. Some subsonic, subsonic-sonic, and transonic flow patterns are investigated. The emphasizes will be on subsonic-sonic flows in a finite convergent nozzle (which is a degenerate free-boundary problem) and a transonic-shock flows in a de Laval nozzle proposed by Courant-Friedrichs. Both well-posedness and ill-posedness results will be presented.
Abstract:
We perform a rigorous analysis of the quasi-neutral limit for a hydrodynamic model of a viscous plasma represented by the Navier-Stokes-Poisson system of equations.
Title: Two-Dimensional Riemann Problems for Compressible Euler System
Abstract:
Riemann problems for the compressible Euler system in two space dimensions remain open, but not intact. In fact, some very interesting progress has been made both numerically and theoretically as well as asymptotically, such as the rigorous result of interaction of binary and some quadruple rarefaction waves and swirling flows. Some new insight are gained into the structure of transonic flow, e.g. von Neumann waves or Guderley waves through patches of the so-called ``semi-hyperbolic'' waves. We try to sketch a rough progress report and outline the next layer of issues. It will be based on work accomplished over the years with Tong Zhang, Jiequan Li, Zhen Lei, K. Song, S. Bang, Mingjie Li, Zhicheng Yang, James Glimm, Xiaomei Ji, et.al.
Title: A Divergence-Free Finite Element Method for the 3D Navier-Stokes Equations in the Vorticity-Vector Potential Form
Abstract:
The vorticity-vector potential formulation of the Navier-Stokes equations is an important alternative to the velocity-pressure formulation, as the vorticity characterizes singularity and turbulence better than the velocity, a lower order primitive variable [6, 3]. Unlike the 2D vorticity-streamline formulation [1, 2, 5, 7], it is a challenge to implement the 3D divergence-free constraint on the vorticity and on the vector potential, computationally. By constructing 3D divergence-free finite element spaces and local basis functions, we can apply a finite element method to the Navier-Stokes equations to obtain $H^1$ vorticity and vector potential solutions which are both divergence-free pointwise. Numerical tests are presented to support the analysis, and are compared to those obtained by a method from [4] where $H(\hbox{curl})$ elements are used instead of $H^1$ divergence-free elements.
References: