Numerical Analysis Archives for Academic Year 2017


Waves in viscoelastic media

When: Tue, September 5, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Francisco-Javier Sayas (Department of Mathematical Sciences, University of Delaware) - http://www.math.udel.edu/~fjsayas/
Abstract: I will explain ongoing work with my team (Tom Brown, Shukai Du, and Hasan Eruslu) on Finite Element simulation of elastic wave propagation in media that are modeled using a strain-to-stress relation that keeps track of the past evolution of the solid. The family of models I will discuss includes the classical (Zener) differential viscoelastic model, a fractional derivative version thereof, and combinations of the above with pure elastic behavior. The analysis will be presented using transfer function techniques, but I will explain how in some cases refined results can be found using techniques of semigroup theory.

Finite element approximation of nonconvex uniformly elliptic fully nonlinear equations

When: Tue, September 12, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Abner J. Salgado (University of Tennessee, Knoxville) - http://www.math.utk.edu/~abnersg/
Abstract: We propose and analyze a two-scale finite element method for the Isaacs equation. By showing the consistency of the approximation and that the method satisfies the discrete maximum principle we establish convergence to the viscosity solution. By properly choosing each of the scales, and using the recently derived discrete Alexandrov Bakelman Pucci estimate we can deduce rates of convergence.

An angular momentum method for approximating wave maps into the sphere

When: Tue, September 19, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Franziska Weber (Department of Mathematics, University of Maryland, College Park) - https://terpconnect.umd.edu/~frweber/
Abstract: We present a convergent finite difference method for approximating wave maps into the sphere. The method is based on a reformulation of the second order wave map equation as a first order system by using the angular momentum as an auxiliary variable. This enables us to preserve the length constraint as well as the energy inherit in the system of equations at the discrete level. The method is shown to converge to a weak solution of the wave map equation as the discretization parameters go to zero. Moreover, it is fast in the sense that O(N log N) operations are required in each time step (where N is the number of grid cells) and a linear CFL-condition is sufficient for stability and convergence. The performance of the method is illustrated by numerical experiments.

The method can be extended to a convergent scheme for the damped wave map equation and the heat map flow. If time permits, I will also discuss possible extensions of the method to applications for liquid crystal dynamics.

An optimal adaptive Fictitious Domain Method

When: Thu, October 5, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Rob Stevenson (University of Amsterdam) - https://staff.fnwi.uva.nl/r.p.stevenson/
Abstract: We consider a Fictitious Domain formulation of an elliptic PDE, and solve the arising saddle-point problem by an inexact preconditioned Uzawa iteration.
Solving the arising `inner' elliptic problems with an adaptive finite element method, we prove that the overall method converges with the best possible rate.
So far our results apply to two-dimensional domains and lowest order finite elements (continuous piecewise linears on the fictitious domain, and piecewise constants on the boundary of the physical domain).

Joint work with S. Berrone (Torino), A. Bonito (Texas A&M), and M. Verani (Milano).

Adaptive Higher-Order Finite Element Simulations for the Time-Harmonic Maxwell Equations

When: Tue, October 10, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Matthias Maier (University of Minnesota) - http://www-users.math.umn.edu/~msmaier/
Abstract: In the terahertz frequency range, the effective (complex-valued) surface conductivity of atomically thick 2D materials such as graphene has a positive imaginary part that is considerably larger than the real part. This feature allows for the propagation of slowly decaying electromagnetic waves, called surface plasmon-polaritons (SPPs), that are confined near the material interface with wavelengths much shorter than the wavelength of the free-space radiation. SPPs are promising ingredients in the design of novel optical applications promising "subwavelength optics" beyond the diffraction limit. There is a compelling need for controllable numerical schemes which, placed on firm mathematical grounds, can reliably describe SPPs in a variety of geometries.

In this talk we present a higher-order finite element approach for the simulation of SPP structures on a conducting sheet, excited by a plane-wave or electric Hertzian dipole sources. Aspects of the numerical treatment such as absorbing, perfectly matched layers, local refinement and a-posteriori error control are discussed. Corresponding analytical results are briefly presented as well.

A Finite Element Scheme for a Phase Field Model of Nematic Liquid Crystal Droplets

When: Tue, October 17, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Shawn Walker (Louisiana State University) - https://www.math.lsu.edu/~walker/
Abstract: We present a phase field model for nematic liquid crystal droplets. Our model couples the Cahn-Hilliard equation to Ericksen's one constant model for liquid crystals with variable degree of orientation. We present a special discretization of the liquid crystal energy that can handle the degenerate elliptic part without regularization. In addition, our discretization uses a mass lumping technique in order to handle the unit length constraint. Discrete minimizers are computed via a discrete gradient flow. We prove that our discrete energy Gamma-converges to the continuous energy and our gradient flow scheme is monotone energy decreasing. Numerical simulations will be shown in 2-D to illustrate the method. This work is joint with Amanda Diegel (post-doc at LSU).

Near the end of the talk, I will discuss 3-D simulations of the Ericksen model coupled to the Allen-Cahn equations (with a mass constraint). This work is joint with REU 2017 students (E. Seal and A. Morvant).

A new and robust approach to construct energy stable schemes for gradient flows

When: Tue, October 24, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Jie Shen (Purdue University) - https://www.math.purdue.edu/~shen/
Abstract: We propose a new technique, the single auxiliary variable (SAV) approach, to deal with nonlinear terms in a large class of gradient flows. The technique is not restricted to specific forms of the nonlinear part of the free energy, it leads to linear and unconditionally energy stable second-order (or higher-order with weak stability conditions) schemes which only require solving decoupled linear equations with constant coefficients. Hence, these schemes are extremely efficient as well as accurate.

We apply the SAV approach to deal with several challenging applications which can not be easily handled by existing approaches, and present convincing numerical results to show that the new schemes are not only much more efficient and easy to implement, but also can better capture the physical properties in these models.

Adaptive FEM for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver

When: Tue, October 31, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Christian Glusa (Sandia National Laboratories) -
Abstract: We explore the connection between fractional order partial differential
equations in two or more spatial dimensions with boundary integral
operators to develop techniques that enable one to efficiently tackle
the integral fractional Laplacian. We develop all of the components
needed to construct an adaptive finite element code that can be used to
approximate fractional partial differential equations, on non-trivial
domains in \(d\geq 1\) dimensions. Our main approach consists of taking
tools that have been shown to be effective for adaptive boundary element
methods and, where necessary, modifying them so that they can be applied
to the fractional PDE case. Improved a priori error estimates are
derived for the case of quasi-uniform meshes which are seen to deliver
sub-optimal rates of convergence owing to the presence of singularities.
Attention is then turned to the development of an a posteriori error
estimate and error indicators which are suitable for driving an adaptive
refinement procedure. We assume that the resulting refined meshes are
locally quasi-uniform and develop efficient methods for the assembly of
the resulting linear algebraic systems and their solution using
iterative methods, including the multigrid method. The storage of the
dense matrices along with efficient techniques for computing the dense
matrix vector products needed for the iterative solution is also
considered. Importantly, the approximation does not make any strong
assumptions on the shape of the underlying domain and does not rely on
any special structure of the matrix that could be exploited by fast
transforms. The performance and efficiency of the resulting algorithm is
illustrated for a variety of examples.

This is joint work with Mark Ainsworth, Brown University.

Optimal Estimation and Computation from Data

When: Tue, November 7, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Simon Foucart (Texas A&M University) - http://www.math.tamu.edu/~foucart/
Abstract: Scientific problems often feature observational data received in the form $w_1=l_1(f), \ldots$, $w_m=l_m(f)$ of known linear functionals applied to an unknown function $f$ from some Banach space $\mathcal{X}$, and it is required to either approximate $f$ (the full approximation problem) or to estimate a quantity of interest $Q(f)$. In typical examples, the quantities of interest can be the maximum/minimum of $f$ or some averaged quantity such as the integral of $f$, while the observational data consists of point evaluations. To obtain meaningful results about such problems, it is necessary to possess additional information about $f$, usually as an assumption that $f$ belongs to a certain model class $\mathcal{K}$ contained in $\mathcal{X}$. This is precisely the framework of optimal recovery, which produced substantial investigations when the model class is a ball of a smoothness space, e.g. when it is a Lipschitz, Sobolev, or Besov class. This presentation concentrates on other model classes described by approximation processes. The main innovations are:
(i) for the estimation of quantities of interest, the production of numerically implementable algorithms which are optimal over these model classes,
(ii) for the full approximation problem, the construction of linear algorithms which are optimal or near optimal over these model classes in case of data consisting of point evaluations.
Regarding (i), when $Q$ is a linear functional, the existence of linear optimal algorithms was established by Smolyak, but the proof was not numerically constructive. In classical recovery settings, it is shown here that such linear optimal algorithms can be produced by constrained minimization methods, and examples involving the computations of integrals from the given data are examined in greater details. Regarding (ii), it is shown that linearization of optimal algorithms can be achieved for the full approximation problem, too, in the important situation where the $l_j$ are point evaluations and $\mathcal{X}$ is a space of continuous functions equipped with the uniform norm. It is also revealed how the quasi-interpolation theory allows for the construction of linear algorithms which are near optimal.

Numerical computations with functions defined on the sphere and disk

When: Tue, November 14, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Alex Townsend (Cornell University) - http://www.math.cornell.edu/~ajt/
Abstract: A classical technique for computing with functions on the sphere and disk is to "double up" the domain, leading to regularity preserving approximants. We synthesize this with new techniques for constructing low rank function approximations to develop a whole collection of fast and adaptive algorithms for sphere and disk computations that are accurate to machine precision. Applications include vector calculus, the solution of PDEs, and the long-time simulation of active biological fluids. This is joint work with Heather Wilber and Grady Wright from Boise State University.

TBA

When: Tue, December 12, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Mert Gurbuzbalaban (Rutgers University) - https://mert.lids.mit.edu