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This study derives geometric, variational discretizations of continuum theories arising in fluid dynamics , magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on… (More)

This study investigates Lagrangian coherent structures (LCS) in the planar elliptic restricted three-body problem (ER3BP), a generalization of the circular restricted three-body problem (CR3BP) that asks for the motion of a test particle in the presence of two elliptically orbiting point masses. Previous studies demonstrate that an understanding of… (More)

We introduce a framework for the design of finite element methods for two-dimensional moving boundary problems with prescribed boundary evolution that have arbitrarily high order of accuracy, both in space and in time. At the core of our approach is the use of a universal mesh: a stationary background mesh containing the domain of interest for all times… (More)

We derive upper bounds on the difference between the orthogonal projections of a smooth function u onto two finite element spaces that are nearby, in the sense that the support of every shape function belonging to one but not both of the spaces is contained in a common region whose measure tends to zero under mesh refinement. The bounds apply, in… (More)

We present a unified analysis of finite element methods for problems with prescribed moving boundaries. In particular, we study an abstract parabolic problem posed on a moving domain with prescribed evolution, discretized in space with a finite element space that is associated with a moving mesh that conforms to the domain at all times. The moving mesh is… (More)

SUMMARY We propose a family of methods for simulating two-dimensional incompressible, low Reynolds number flow around a moving obstacle whose motion is prescribed. The methods make use of a universal mesh: a fixed background mesh that adapts to the geometry of the immersed obstacle at all times by adjusting a few elements in the neighborhood of the… (More)

- Mathieu Desbrun, Evan S. Gawlik, François Gay-Balmaz, Vladimir Zeitlin
- 2013

In this paper we develop and test a structure-preserving discretiza-tion scheme for rotating and/or stratified fluid dynamics. The numerical scheme is based on a finite dimensional approximation of the group of volume preserving diffeomorphisms recently proposed in [25, 9] and is derived via a discrete version of the Euler-Poincaré variational formulation… (More)

With an environment comparable to that of primordial Earth, a surface strewn with liquid hydrocarbon lakes, and an atmosphere denser than that of any other moon in the solar system, Saturn's largest moon Titan is a treasure trove of potential scientific discovery and is the target of a proposed NASA mission scheduled for launch in roughly one decade. A… (More)

We derive iterative methods for computing the Fréchet derivative of the map which sends a full-rank matrix A to the factor U in its polar decomposition A = U H, where U has orthonormal columns and H is Hermitian positive definite. The methods apply to square matrices as well as rectangular matrices having more rows than columns. Our derivation relies on a… (More)

We study schemes for interpolating functions that take values in the special orthogonal group SO(n). Our focus is on interpolation schemes obtained by embedding SO(n) in a linear space, interpolating in the linear space, and mapping the result onto SO(n) via the closest point projection. The resulting interpolants inherit both the order of accuracy and the… (More)