Thu, 15 Nov 2012

14:00 - 15:00
Gibson Grd floor SR

Optimally Blended Spectral-Finite Element Scheme for Wave Propagation and Non-Standard Reduced Integration

Professor Mark Ainsworth
(Brown University)
Abstract

We study the dispersion and dissipation of the numerical scheme obtained by taking a weighted averaging of the consistent (finite element) mass matrix and lumped (spectral element) mass matrix for the small wave number limit. We find and prove that for the optimum blending the resulting scheme

(a) provides $2p+4$ order accuracy for $p$th order method (two orders more accurate compared with finite and spectral element schemes);

(b) has an absolute accuracy which is $\mathcal{O}(p^{-3})$ and $\mathcal{O}(p^{-2})$ times better than that of the pure finite and spectral element schemes, respectively;

(c) tends to exhibit phase lag.

Moreover, we show that the optimally blended scheme can be efficiently implemented merely by replacing the usual Gaussian quadrature rule used to assemble the mass and stiffness matrices by novel nonstandard quadrature rules which are also derived.

Mon, 17 Jan 2011

17:00 - 18:00
Gibson 1st Floor SR

Linear instability of the Relativistic Vlasov-Maxwell system

Jonathan Ben-Artzi
(Brown University)
Abstract

We consider the Relativistic Vlasov-Maxwell system of equations which

describes the evolution of a collisionless plasma. We show that under

rather general conditions, one can test for linear instability by

checking the spectral properties of Schrodinger-type operators that

act only on the spatial variable, not the full phase space. This

extends previous results that show linear and nonlinear stability and

instability in more restrictive settings.

Tue, 26 May 2009

15:45 - 16:45
L3

Gluing constructions of special Lagrangian cones

Nicos Kapouleas
(Brown University)
Abstract

I will survey the recent work of Haskins and myself constructing new special Lagrangian cones in ${\mathbb C}^n$

for all $n\ge3$ by gluing methods. The link (intersection with the unit sphere ${\cal S}^{2n-1}$) of a special Lagrangian cone is a special Legendrian $(n-1)$-submanifold. I will start by reviewing the geometry of the building blocks used. They are rotationally invariant under the action of $SO(p)\times SO(q)$ ($p+q=n$) special Legendrian $(n-1)$-submanifolds of ${\cal S}^{2n-1}$. These we fuse (when $p=1$, $p=q$) to obtain more complicated topologies. The submanifolds obtained are perturbed to satisfy the special Legendrian condition (and their cones therefore the special Lagrangian condition) by solving the relevant PDE. This involves understanding the linearized operator and its small eigenvalues, and also ensuring appropriate decay for the solutions.

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