TAMU Seminars in Applied Mathematics and Scientific Computation

Abstract: We derive a posteriori error estimates for time discrete approximations for Schroedinger-type and semilinear parabolic equations. In the case of linear Schroedinger equations fully discrete approximations will be studied as well. The estimates are of optimal order and aim to provide information for the behavior of the method even in cases where we do not have information on the exact solution of the p.d.e. In the nonlinear cases we will mainly consider problems with possible blow-up in finite time. In these cases we derive conditional estimates (estimates which are valid under conditions of a posteriori type). We further discuss how these estimates can lead to error control near the blow-up time.

Host: Andrea Bonito

Abstract: The subsequent discretization of the EFIE is a common approach to solve scattering problems on unbounded domains which is known as the Boundary Element Method (BEM) or Method of Moments (Mom). In many applications, such as optimization, shape recognition or inverse problems, just to mention a few, solving the Boundary Element Method for each new parameter value is too expensive (and unnecessary).

The Reduced Basis Method is accurate, efficient and trustable algorithm in the framework of parametrized problems and in a many-query context. We will present how the Reduced Basis Method is applied to parametrized scattering problems. The novelty is that for the first time the Reduced Basis Method is applied to an integral equation. We will discuss the challenges and present numerical examples.

Title: Stability of Oscillating Pipe Flow

Title: TBA

Abstract: In this talk, we report some recent works on the diffuse interface modeling and simulation of interface problems in materials science and biology. Particular examples include the studies of biomimetic vesicle membranes and homogeneous nucleation in anisotropic elastic solids. In both cases, elastic energy contributions are taken into account. We consider various theoretical and computational issues related to diffuse interface models and present some simulations results. We also discuss how to connect geometry, topology and analysis closely within a diffuse interface framework.

Abstract: We have calculated a six-dimensional ab initio potential energy surface for the OC-HF dimer at the CCSD(T)/aug-cc-pVTZ level of theory. A least-squares fitting method was used to fit the angular part of the potential, and the six-dimensional surface was obtained by interpolating the angular potential on a grid of R, rCO, and rHF points, using a three-dimensional reproducing kernel Hilbert spaces. In order to reduce the dimensionality of the system from six to four dimensions, the C-O and H-F stretching motions were adiabatically separated from the bending and stretching motions of the complex, using the vibrational self-consistent field method. The Hamiltonian operator has been split into six different terms, where the kinetic energy operators are diagonal in the spectral representation of the wave function. On the other hand, the potential energy is diagonal in the grid representation of the wave function. The spectral and grid representation are related through a unitary transformation. The total representation of the Hamiltonian operator in the spectral basis is obtained by transforming the vector in the grid representation to a spectral representation. Then the rovibrational energy levels were calculated using a Lanczos based iterative diagonalization scheme. This scheme requires repetitively acting the Hamiltonian operator on a vector, while avoiding the problem of constructing the full matrix.

Abstract: We consider time-dependent, incompressible flow problems discretized

via higher order finite element methods. Applying an IMEX scheme or a fully implicit time discretization and a linearization method leads to a saddle point system. This linear system is solved using a preconditioned Krylow method, which is fully parallelized on a distributed memory parallel computer.

We introduce a robust block-triangular preconditioner and beside numerical results of the parallel performance we explain and evaluate the main building blocks of the parallel implementation.

Equations

Abstract:

In many practical applications, for instance, in computational electromagnetics, the excitation is time-harmonic. Switching from the time domain to the frequency domain allows us to avoid expensive time-stepping schemes by solving a simple elliptic equation for the amplitude. This is possible for linear problems, but not for nonlinear problems. However, due to the periodicity of the solution, we can expand the solution in a Fourier series. Truncating this Fourier series and approximating the Fourier coefficients by finite elements, we arrive at a large-scale coupled nonlinear system for determining the finite element approximation to the Fourier coefficients. The construction of fast solvers for such systems is very crucial for the efficiency of this multiharmonic approach. In this talk, we construct and analyze nearly optimal solvers for the Jacobi systems arising from the Newton linearization of the large-scale coupled nonlinear system. Numerical experiments demonstrate the performance of the solver.

Abstract:

For problems exhibiting strong anisotropic features, e.g., boundary

and internal layers in fluid mechanics and aerodynamics, flame fronts

in combustion, and phase boundaries in heat transfer and material

processing, finite element methods baseds on adaptive anisotropic

meshes can be much more effective than those on isotropic meshes.

In this talk, we present our resent results on the error estimates

and mesh refinement controls for anisotropic FEM. We first introduce

some measures to characterize the anisotropic features of the higher

order derivatives of a function, and derive the error estimates for

the piecewise polynomial interpolations on anisotropic meshes. Then

we describe some recovery techniques in FEM for constructing an

approximation of the higher order derivatives of the PDE solutions.

Based on the recovered derivatives, we are able to design certain

mesh metrics for guiding the anisotropic mesh refinement process.

Numerical results demonstrate that such designed adaptive FEM

maintains the optimal convergence rate in the case of anisotropic

meshes with high aspect ratios.

Abstract: In the context of fluid flow stable and non-dissipative time-stepping schemes are required. In this talk a new variant of the fractional-step time-stepping scheme which was proposed by Glowinski is analyzed. The numerical stability, dissipation, and damping properties are considered and compared to other well known time-stepping schemes. These criteria are discussed for choosing appropriate time-stepping schemes for longtime computations in fluid flow. The various schemes were considered by the means of numerical examples.

three-body interactions.

Abstract: In this talk we will discuss the dynamics of a boson gas with three-body interactions in dimensions d=1,2. We prove that in the limit where the particle number N tends to infinity, the BBGKY hierarchy of k-particle marginals converges to a limiting Gross-Pitaevskii (GP) hierarchy for which we prove existence and uniqueness of solutions. For factorized initial data, the solutions of the GP hierarchy are shown to be determined by solutions of a quintic nonlinear Schr\"odinger equation.

Time permitting, we will discuss the new approach to look at the the Cauchy problem for focusing and defocusing GP hierarchy. In that line of work we consider the GP hierarchy in dimensions d\geq 1, for cubic, quintic, focusing and defocusing interactions. We prove local in time existence and uniqueness of solutions in generalized Sobolev spaces of sequences of marginal density matrices. This result includes the proof of an a priori spacetime bound conjectured by Klainerman and Machedon for the cubic GP hierarchy in d=3. For defocusing interactions, we prove the existence and uniqueness of solutions globally in time in certain cases. For the focusing GP hierarchies, we prove lower bounds on the blowup rate. These results hold without the assumption of factorized initial conditions.

Host: Guermond

Abstract: We present a new element for a mixed formulation of elasticity with weakly imposed stress symmetry. In many ways this element can be thought of as the elasticity analogue of the Raviart-Thomas element for the Laplace equation.

Abstract: We present review of remapping methods for Arbitrary

Lagrangian-Eulerian (ALE) methods. We first consider

different approaches for remapping of cell-centered

quantities. It includes methods based on exact

intersection (overlays), swept region integration,

flux-corrected remapping, and error compensation.

Then we describe remapping of nodal quantities like

velocity. Finally, we present remapping for multimaterial case.

Title: TBA