17,947 research outputs found
Numerical homogenization of H(curl)-problems
If an elliptic differential operator associated with an
-problem involves rough (rapidly varying)
coefficients, then solutions to the corresponding
-problem admit typically very low regularity, which
leads to arbitrarily bad convergence rates for conventional numerical schemes.
The goal of this paper is to show that the missing regularity can be
compensated through a corrector operator. More precisely, we consider the
lowest order N\'ed\'elec finite element space and show the existence of a
linear corrector operator with four central properties: it is computable,
-stable, quasi-local and allows for a correction of
coarse finite element functions so that first-order estimates (in terms of the
coarse mesh-size) in the norm are obtained provided
the right-hand side belongs to . With these four
properties, a practical application is to construct generalized finite element
spaces which can be straightforwardly used in a Galerkin method. In particular,
this characterizes a homogenized solution and a first order corrector,
including corresponding quantitative error estimates without the requirement of
scale separation
Divergent Perturbation Series
Various perturbation series are factorially divergent. The behavior of their
high-order terms can be found by Lipatov's method, according to which they are
determined by the saddle-point configurations (instantons) of appropriate
functional integrals. When the Lipatov asymptotics is known and several lowest
order terms of the perturbation series are found by direct calculation of
diagrams, one can gain insight into the behavior of the remaining terms of the
series. Summing it, one can solve (in a certain approximation) various
strong-coupling problems. This approach is demonstrated by determining the
Gell-Mann - Low functions in \phi^4 theory, QED, and QCD for arbitrary coupling
constants. An overview of the mathematical theory of divergent series is
presented, and interpretation of perturbation series is discussed. Explicit
derivations of the Lipatov asymptotic forms are presented for some basic
problems in theoretical physics. A solution is proposed to the problem of
renormalon contributions, which hampered progress in this field in the late
1970s. Practical schemes for summation of perturbation series are described for
a coupling constant of order unity and in the strong-coupling limit. An
interpretation of the Borel integral is given for 'non-Borel-summable' series.
High-order corrections to the Lipatov asymptotics are discussed.Comment: Review article, 45 pages, PD
Using hybrid GPU/CPU kernel splitting to accelerate spherical convolutions
We present a general method for accelerating by more than an order of
magnitude the convolution of pixelated functions on the sphere with a
radially-symmetric kernel. Our method splits the kernel into a compact
real-space component and a compact spherical harmonic space component. These
components can then be convolved in parallel using an inexpensive commodity GPU
and a CPU. We provide models for the computational cost of both real-space and
Fourier space convolutions and an estimate for the approximation error. Using
these models we can determine the optimum split that minimizes the wall clock
time for the convolution while satisfying the desired error bounds. We apply
this technique to the problem of simulating a cosmic microwave background (CMB)
anisotropy sky map at the resolution typical of the high resolution maps
produced by the Planck mission. For the main Planck CMB science channels we
achieve a speedup of over a factor of ten, assuming an acceptable fractional
rms error of order 1.e-5 in the power spectrum of the output map.Comment: 9 pages, 11 figures, 1 table, accepted by Astronomy & Computing w/
minor revisions. arXiv admin note: substantial text overlap with
arXiv:1211.355
An extended finite element method with smooth nodal stress
The enrichment formulation of double-interpolation finite element method
(DFEM) is developed in this paper. DFEM is first proposed by Zheng \emph{et al}
(2011) and it requires two stages of interpolation to construct the trial
function. The first stage of interpolation is the same as the standard finite
element interpolation. Then the interpolation is reproduced by an additional
procedure using the nodal values and nodal gradients which are derived from the
first stage as interpolants. The re-constructed trial functions are now able to
produce continuous nodal gradients, smooth nodal stress without post-processing
and higher order basis without increasing the total degrees of freedom. Several
benchmark numerical examples are performed to investigate accuracy and
efficiency of DFEM and enriched DFEM. When compared with standard FEM,
super-convergence rate and better accuracy are obtained by DFEM. For the
numerical simulation of crack propagation, better accuracy is obtained in the
evaluation of displacement norm, energy norm and the stress intensity factor
Well-posedness of Multidimensional Diffusion Processes with Weakly Differentiable Coefficients
We investigate well-posedness for martingale solutions of stochastic
differential equations, under low regularity assumptions on their coefficients,
widely extending some results first obtained by A. Figalli. Our main results
are a very general equivalence between different descriptions for
multidimensional diffusion processes, such as Fokker-Planck equations and
martingale problems, under minimal regularity and integrability assumptions,
and new existence and uniqueness results for diffusions having weakly
differentiable coefficients, by means of energy estimates and commutator
inequalities. Our approach relies upon techniques recently developed, jointly
with L. Ambrosio, to address well-posedness for ordinary differential equations
in metric measure spaces: in particular, we employ in a systematic way new
representations and inequalities for commutators between smoothing operators
and diffusion generators.Comment: Added references to further literature on the subjec
3D cut-cell modelling for high-resolution atmospheric simulations
Owing to the recent, rapid development of computer technology, the resolution
of atmospheric numerical models has increased substantially. With the use of
next-generation supercomputers, atmospheric simulations using horizontal grid
intervals of O(100) m or less will gain popularity. At such high resolution
more of the steep gradients in mountainous terrain will be resolved, which may
result in large truncation errors in those models using terrain-following
coordinates. In this study, a new 3D Cartesian coordinate non-hydrostatic
atmospheric model is developed. A cut-cell representation of topography based
on finite-volume discretization is combined with a cell-merging approach, in
which small cut-cells are merged with neighboring cells either vertically or
horizontally. In addition, a block-structured mesh-refinement technique is
introduced to achieve a variable resolution on the model grid with the finest
resolution occurring close to the terrain surface. The model successfully
reproduces a flow over a 3D bell-shaped hill that shows a good agreement with
the flow predicted by the linear theory. The ability of the model to simulate
flows over steep terrain is demonstrated using a hemisphere-shaped hill where
the maximum slope angle is resolved at 71 degrees. The advantage of a locally
refined grid around a 3D hill, with cut-cells at the terrain surface, is also
demonstrated using the hemisphere-shaped hill. The model reproduces smooth
mountain waves propagating over varying grid resolution without introducing
large errors associated with the change of mesh resolution. At the same time,
the model shows a good scalability on a locally refined grid with the use of
OpenMP.Comment: 19 pages, 16 figures. Revised version, accepted for publication in
QJRM
Boundary regularity for elliptic systems under a natural growth condition
We consider weak solutions of second order nonlinear elliptic systems of the type
in with an inhomogeneity
satisfying a natural growth condition. In dimensions we show
that -almost every boundary point is a regular point for
, provided that the boundary data and the coefficients are sufficiently
smooth.Comment: revised version, accepted for publication in Ann. Mat. Pura App
Analysis of a high order Trace Finite Element Method for PDEs on level set surfaces
We present a new high order finite element method for the discretization of
partial differential equations on stationary smooth surfaces which are
implicitly described as the zero level of a level set function. The
discretization is based on a trace finite element technique. The higher
discretization accuracy is obtained by using an isoparametric mapping of the
volume mesh, based on the level set function, as introduced in [C. Lehrenfeld,
\emph{High order unfitted finite element methods on level set domains using
isoparametric mappings}, Comp. Meth. Appl. Mech. Engrg. 2016]. The resulting
trace finite element method is easy to implement. We present an error analysis
of this method and derive optimal order -norm error bounds. A
second topic of this paper is a unified analysis of several stabilization
methods for trace finite element methods. Only a stabilization method which is
based on adding an anisotropic diffusion in the volume mesh is able to control
the condition number of the stiffness matrix also for the case of higher order
discretizations. Results of numerical experiments are included which confirm
the theoretical findings on optimal order discretization errors and uniformly
bounded condition numbers.Comment: 28 pages, 5 figures, 1 tabl
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