Title: Shape deformations and analytic continuation in free boundary problems
Abstract:
In this talk I will present a general approach to the analysis of
steady states and nonlinear stability for a class of free-boundary
problems. The approach is based on the explicit consideration of
series solutions in a parameter measuring variations of domains
(e.g. steady state or initial configurations) from separable
geometries (e.g. planes, spheres, etc). The method relies on the
derivation of formulas for the recursive evaluation of (Taylor)
approximations of arbitrary order in the variation parameters, and
on the iterative estimation of the growth, in appropriately defined
spaces, of the resulting functional coefficients. We shall present
a variety of instances where this procedure leads to general results
on analyticity of solutions, on existence of complex steady states
and on nonlinear stability of equilibria. Particular attention will
be given to examples of "curvature driven" free-boundary problems,
including the classical Stefan problems, models of capillary fluid
drops, water waves, tumor growth, etc. We shall further show that
the relevance of our studies goes beyond the theoretical, as it
uncovered the mechanism behind the observed performance of a class
of numerical algorithms, based on shape-perturbation theory, that
have been used in these and other contexts. Indeed, our research
shows that the standard implementations of these schemes can suffer
from severe ill-conditioning as a result of pronounced cancellations
in the underlying recursions. Moreover, our work further suggests
alternative implementations with greatly improved properties of
numerical stability and convergence, enhanced by methods of analytic
continuation. As we shall demonstrate these modifications can have
a dramatic effect on the accuracy and applicability of perturbative
numerical approaches to (boundary value and) free-boundary problems.
Title: Multilevel-Preconditioning for Discontinuous Galerkin Methods
Abstract:
Variants of a multi-level preconditioner for the interior penalty
method are presented. They are analyzed in the framework established
by Bramble/Pasciak for symmetric problems and shown to be optimal.
Then, the methods are extended to other discontinuous discretizations
of elliptic problems. Finally, they are applied to problems of
computational fluid dynamics.
Title: Mixed hp-DGFEM for incompressible flows
Abstract:
We consider the discretization of incompressible fluid
flow problems by mixed discontinuous Galerkin methods.
The derivation of these methods is explained for
velocity-pressure elements of the type ${\cal Q}_k-{\cal Q}_k$
and ${\cal Q}_k-{\cal Q}_{k-1}$. It is shown that these
elements satisfy suitable inf-sup conditions on geometric
meshes in polygonal and polyhedral domains. The meshes may
be refined anisotropically and non quasi-uniformly
towards edges and corners, respectively. The discrete
inf-sup constants are proven to be independent of the
aspect ratio of the anisotropic elements and only weakly
depending on the approximation orders. Using these results,
we derive a-priori error estimates for $hp$-approximations
on geometric meshes. A series of numerical tests is
presented for Stokes and Oseen flow. In particular,
the results demonstrate that the methods perform well
for a wide range of the Reynolds numbers.
Title: A new characterization of hybridized mixed methods
Abstract:
Mixed finite element methods are a very powerful method for numerically
solving second-order elliptic problems. Since their approximate solution
is difficult to compute, it is necessary to recast them in a suitable way;
this can be achieved by what is called a hybridization procedure. The
price to pay, however, is that we must introduce new unknowns, called
Lagrange multipliers, which satisfy a matrix equation difficult to
compute. In this talk, we give a new characterization of the
approximate solution given by hybridized mixed methods for second-order,
self-adjoint elliptic problems. We then apply this characterization to
obtain an explicit formula for the entries of the matrix equation for the
so-called Lagrange multipliers. We also obtain necessary and sufficient
conditions under which the multipliers of two well-known methods (the
Raviart-Thomas and the Brezzi-Douglas-Marini) of similar order are
identical.
Title: A new characterization of hybridized mixed methods
Abstract:
Mixed finite element methods are a very powerful method for numerically
solving second-order elliptic problems. Since their approximate solution
is difficult to compute, it is necessary to recast them in a suitable way;
this can be achieved by what is called a hybridization procedure. The
price to pay, however, is that we must introduce new unknowns, called
Lagrange multipliers, which satisfy a matrix equation difficult to
compute. In this talk, we give a new characterization of the
approximate solution given by hybridized mixed methods for second-order,
self-adjoint elliptic problems. We then apply this characterization to
obtain an explicit formula for the entries of the matrix equation for the
so-called Lagrange multipliers. We also obtain necessary and sufficient
conditions under which the multipliers of two well-known methods (the
Raviart-Thomas and the Brezzi-Douglas-Marini) of similar order are
identical.
Title: Modeling flows of nematic liquid crystalline polymers using kinetic theory
Abstract:
I will present a general Doi-type kinetic theory for nematic liquid crystalline polymers (LCPs) accounting for
the molecular aspect ratio, excluded volume interaction, long-range intermolecular interaction,
and chirality of the molecules. I will derive an approximate intermolecular potential for LCP
molecules of the spheroidal shape and the elastic as well as the visocus stress expression
corresponding to the shaped molecule being transported in viscous solvent. I will then show the theory
obeys the second law of thermodynamics and "reduces" to the well-known Ericksen-Leslie theory for
nematic liquid crystals in the weak flow, weak elasticity, and slow time limit. Applications of
the theory and its moment approximations in simple flows will be discussed in the end.
Title: Hyperbolic formulations in linearized gravity
Abstract:
The problem of running stable evolution for the Einstein's equation
keeps challenging scientists for many years. With the development
of Laser Interferometer Gravitational Observatories whose
main purpose is the detection of the gravity waves, the need in reliable
computations had grown tremendously. At the same time, difficulties
of numerical relativity are so intense and various that one can hardly
expect a real success without the deepest analysis of the subject.
What are the challenges that make the discretization of the
relativity equations difficult? Possibly, many questions will drop
out once we know a natural way of writing the Einstein's equation.
Indeed, in terms of curvature and stress-energy tensors the Einstein's
equation is short, but when translated into a coordinate system, it
turns into a system of 10 nonlinear PDE's with a huge amount of terms,
and from 10 unknown components of the space-time metric only 6 can be
determined.
Due to this one and due to other reasons which will be addressed in
the talk various reformulation were proposed to write the
Einstein's equation in a manner suitable for numeric computations.
The most promising approach is to write the Einstein equation in the form
of a symmetric hyperbolic system. In this talk we will discuss examples of
well-known symmetric hyperbolic formulations and introduce a new one
which is the result of our work with Professor Arnold during the last
year. Finally, we will briefly mention the future challenges
in particular, the admissible boundary conditions and approaches to
discretization.
Title: Numerical simulation of drop coalescence with surfactant in 3D
Abstract:
In the processing of emulsions and polymer blends, the drop size
distributions are determined by two coexisting processes: drop breakup
and coalescence. Here we study the effects of surfactants, e.g. block
copolymers, on coalescence. We use a newly developed 3D adaptive
finite-element algorithm. The method is based on unstructured adaptive triangulated and
tetrahedral meshes that discretize the interfaces and the bulk respectively,
and on an efficient parallelization of the numerical solvers.
A nonlinear Langmuir equation of state for the surfactant is used and
Van der Waals forces, which are responsible for coalescence, are included
in the numerical method. Surfactants are transported by convection-diffusion on the
drop/matrix interface and between the interface and the bulk
phases. Our accurate and robust numerical method features parallel
computation and adaptive reconstruction of the finite element meshes
describing the bulk phases and the interface.
Our results reveal a nontrivial dependence of the critical
capillary number Ca_c, below which coalescence occurs,
on the surface coverage of surfactant. Marangoni stresses inhibit
coalescence and thus decrease Ca_c with respect to
the clean-drop case. However, at large surfactant coverages
close to the maximum packing of surfactant molecules,
surfactant redistribution is prohibited (the surfactant is
nearly incompressible) and thus the effect of
Marangoni stresses is weakened, leading to an increase of Ca_c.
In some cases, Ca_c at high coverages is even higher
than in the clean-drop case: surfactant near-incompressibility renders
the interface more rigid which results in less drop deformation and thus
coalescence can occur at higher capillary number.
Finally, our results also reveal a nontrivial dependence of Ca_c
on surfactant solubility in the bulk. At moderate surfactant concentration,
diffusion in the bulk decreases surfactant redistribution
on the interface and thus weakens Marangoni stresses resulting
in higher Ca_c than in the insoluble case. However, when the
surfactant bulk concentration is large, high adsorption
fluxes maintain a higher surface concentration in equilibrium than for
the insoluble case, thus resulting in larger drop deformation and in
lower Ca_c.
This work is joint with H. Zhou, V. Cristini and C.W. Macosko.
Title: Methods for finding singular solutions to nonlinear problems
Abstract:
JOINT COMPUTATIONAL OPTIMIZATION SEMINAR
LOCATED THIS DAY IN LINDH 409.
We consider nonlinear problems for which the Jacobian matrix is singular
at the solution. A new approach for finding singular solutions to
nonlinear equations will be presented. The main idea of this approach is
to replace the original nonlinear equation with a new one having the
same solution as its nonsingular (regular) solution. To construct the
new equation, the information about the first derivative of the original
nonlinear problem is used. The results in this field find wide
applications in bifurcation theory and optimization. An application of
this approach to solving the Chandrasekhar $H$--equation and
reformulations of the nonlinear complementarity problems will be
illustrated.
Title: Evolution of planetary orbits. Hidden symmetry and its relevance to orbit integration.
Abstract:
We revisit the system of equations, which describes orbit evolution
in the celestial mechanics, and point out a previously neglected
aspect of these equations. This system was first written down by
Lagrange, and later rewritten, in a canonical form, by Delaunay. A
careful re-examination of the derivation of this system shows that,
both in the Lagrange and Delaunay formulations, the orbit resides
on a certain 9-dimensional submanifold of some 12-dimensional space
We show that there exists a certain amount of freedom in choosing
the submanifold. This choice is mathematically equivalent to gauge
fixing. The freedom of choice (=freedom of gauge fixing) reveals a
symmetry hiding behind the equations of celestial mechanics. This
symmetry is analogous to the gauge invariance in electrodynamics
(and reveals a fibre-bundle-type structure). Just as a convenient
choice of gauge simplifies calculations in electrodynamics, the
freedom of choice of the submanifold may, potentially, be used to
create simpler schemes of orbit integration. On the other hand, the
presence of this feature may be a previously unrecognised source of
numerical error.
Title: Merging and splitting for scattering calculations
Abstract:
We will show how merging subscatterers is useful for rapid solution
of forward scattering problem, and present an approach to solution of
the inverse problem based on splitting a scatterer into subscatterers.
This splitting process partitions the ill-posed, nonlinear, inverse
problem into a problem that is nonlinear but well-posed, and a problem
that is ill-posed but linear.
Title: Some iterative techniques for time harmonic Maxwell equations
Abstract:
Time harmonic Maxwell equations in a lossless cavity lead to a second
order partial differential equation for electric field involving a
differential operator that is neither elliptic nor definite. A Galerkin
method using Nedelec spaces can be employed to get approximate solutions
numerically. In this talk, we will discuss the problem of efficient
solution of the indefinite linear system arising from this method. Not all
standard techniques work well for this application due to complications
arising from the non-ellipticity and indefiniteness of the underlying
differential operator. The suitability of some Schwarz methods and
multigrid methods for the Maxwell application will be shown. Motivated by
certain analytical techniques developed in this analysis of iterative
techniques, some recent efforts have been made in simplifying the
convergence analysis of finite element methods for time harmonic Maxwell
equations. These will be briefly discussed.
Title: No seminar. Thanksgiving Holiday.
Abstract:
Title: Some Mathematical Issues in Liquid Crystal Flows
Abstract:
We will discuss Ericksen's model of liquid crystals with its
variable degree of orientation. The model consists of governing equations
for the velocity field, the pressure, the director and the order
parameter. The consitutive functions for the Leslie coefficients, derived
from the molecular theory of Doi, play a crucial role in the modeling. One
of the goals of the analysis is to examine the role of the order parameter
in describing defects as well as in obtaining new regimes which cannot be
predicted by the previous Leslie-Ericksen model.
Title: Revenue Management and Markov Decision Processes
Abstract: JOINT COMPUTATIONAL OPTIMIZATION SEMINAR (LINDH409). The talk will begin with a description of the basic revenue management problem, in which a firm must dynamically manage the availability of a variety of products that can be sold for different prices to different customer classes and that are constructed out of a common pool of scarce resources. One important example of such a problem is the fare class management problem faced by commercial airlines, where the resources are seats on flight legs, and the products are possible fare-class/itinerary ticket combinations. The remainder of the talk will describe progress in analyzing these issues using Markov decision processes.