9,530 research outputs found
Optimal Local Multi-scale Basis Functions for Linear Elliptic Equations with Rough Coefficient
This paper addresses a multi-scale finite element method for second order
linear elliptic equations with arbitrarily rough coefficient. We propose a
local oversampling method to construct basis functions that have optimal local
approximation property. Our methodology is based on the compactness of the
solution operator restricted on local regions of the spatial domain, and does
not depend on any scale-separation or periodicity assumption of the
coefficient. We focus on a special type of basis functions that are harmonic on
each element and have optimal approximation property. We first reduce our
problem to approximating the trace of the solution space on each edge of the
underlying mesh, and then achieve this goal through the singular value
decomposition of an oversampling operator. Rigorous error estimates can be
obtained through thresholding in constructing the basis functions. Numerical
results for several problems with multiple spatial scales and high contrast
inclusions are presented to demonstrate the compactness of the local solution
space and the capacity of our method in identifying and exploiting this compact
structure to achieve computational savings
Self-similar Singularity of a 1D Model for the 3D Axisymmetric Euler Equations
We investigate the self-similar singularity of a 1D model for the 3D
axisymmetric Euler equations, which is motivated by a particular singularity
formation scenario observed in numerical computation. We prove the existence of
a discrete family of self-similar profiles for this model and analyze their
far-field properties. The self-similar profiles we find agree with direct
simulation of the model and seem to have some stability
Dynamic Stability of the 3D Axi-symmetric Navier-Stokes Equations with Swirl
In this paper, we study the dynamic stability of the 3D axisymmetric
Navier-Stokes Equations with swirl. To this purpose, we propose a new
one-dimensional (1D) model which approximates the Navier-Stokes equations along
the symmetry axis. An important property of this 1D model is that one can
construct from its solutions a family of exact solutions of the 3D
Navier-Stokes equations. The nonlinear structure of the 1D model has some very
interesting properties. On one hand, it can lead to tremendous dynamic growth
of the solution within a short time. On the other hand, it has a surprising
dynamic depletion mechanism that prevents the solution from blowing up in
finite time. By exploiting this special nonlinear structure, we prove the
global regularity of the 3D Navier-Stokes equations for a family of initial
data, whose solutions can lead to large dynamic growth, but yet have global
smooth solutions
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