599 research outputs found
A note on properties of the restriction operator on Sobolev spaces
No full textIn our companion paper (S.N. Chandler Wilde, D.P. Hewett, A. Moiola, Sobolev
spaces on non-Lipschitz subsets of with application to boundary
integral equations on fractal screens, 2016) we studied a number of different
Sobolev spaces on a general (non-Lipschitz) open subset of
, defined as closed subspaces of the classical Bessel potential
spaces for . These spaces are mapped by the
restriction operator to certain spaces of distributions on . In this
note we make some observations about the relation between these spaces of
global and local distributions. In particular, we study conditions under which
the restriction operator is or is not injective, surjective and isometric
between given pairs of spaces. We also provide an explicit formula for minimal
norm extension (an inverse of the restriction operator in appropriate spaces)
in a special case.Comment: 8 page
On polynomial Trefftz spaces for the linear time-dependent Schr\"odinger equation
No full textWe study the approximation properties of complex-valued polynomial Trefftz
spaces for the -dimensional linear time-dependent Schr\"odinger
equation. More precisely, we prove that for the space-time Trefftz
discontinuous Galerkin variational formulation proposed by G\'omez, Moiola
(SIAM. J. Num. Anal. 60(2): 688-714, 2022), the same -convergence rates as
for polynomials of degree in variables can be obtained in a
mesh-dependent norm by using a space of Trefftz polynomials of anisotropic
degree. For such a space, the dimension is equal to that of the space of
polynomials of degree in variables, and bases are easily constructed
Regularity and asymptotics for p-Laplace type operators in fractal and pre-fractal domains
No full textIn this thesis we deal with double obstacle problems involving p-Laplace type operators in fractal and pre-fractal domains of R^2.
The first improvement here presented consists in establishing a regularity result for the solution to double obstacle problem in terms of weighted Sobolev spaces (involving second derivatives), where we take as weight the distance from the vertex of the reentrant corner.
In particular, we prove local estimates, estimates far away from the conical point and the boundedness of the gradient far away from the conical point.
In addition, thanks to the regularity result presented, we establish a sharp error estimates for the FEM approximations follow the approach of P. Grisvard, considering a suitable triangulation of the domains, adapted to the regularity of the solutions. Furthermore, after stating this optimal estimate, we perform numerical simulations investigating both the cases of p = 2 and p > 2 (fixed).
In the last part of the work, considering both the case of fractal domains and the n-th pre-fractal ones, we investigate the asymptotic behaviour of the solution with respect to p and n. Moreover, we briefly discuss the issue of the uniqueness
Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions
No full textWe consider the Helmholtz transmission problem with one penetrable
star-shaped Lipschitz obstacle. Under a natural assumption about the ratio of
the wavenumbers, we prove bounds on the solution in terms of the data, with
these bounds explicit in all parameters. In particular, the (weighted)
norm of the solution is bounded by the norm of the source term,
independently of the wavenumber. These bounds then imply the existence of a
resonance-free strip beneath the real axis. The main novelty is that the only
comparable results currently in the literature are for smooth, convex obstacles
with strictly positive curvature, while here we assume only Lipschitz
regularity and star-shapedness with respect to a point. Furthermore, our bounds
are obtained using identities first introduced by Morawetz (essentially
integration by parts), whereas the existing bounds use the much-more
sophisticated technology of microlocal analysis and propagation of
singularities. We also recap existing results that show that if the assumption
on the wavenumbers is lifted, then no bound with polynomial dependence on the
wavenumber is possible.Comment: 26 pages, 2 figure
Numerical Quadrature for Singular Integrals on Fractals
No full textWe present and analyse numerical quadrature rules for evaluating regular and
singular integrals on self-similar fractal sets. The integration domain
is assumed to be the compact attractor of an iterated function
system of contracting similarities satisfying the open set condition.
Integration is with respect to any ``invariant'' (also known as ``balanced'' or
``self-similar'') measure supported on , including in particular the
Hausdorff measure restricted to , where is the
Hausdorff dimension of . Both single and double integrals are
considered. Our focus is on composite quadrature rules in which integrals over
are decomposed into sums of integrals over suitable partitions of
into self-similar subsets. For certain singular integrands of
logarithmic or algebraic type we show how in the context of such a partitioning
the invariance property of the measure can be exploited to express the singular
integral exactly in terms of regular integrals. For the evaluation of these
regular integrals we adopt a composite barycentre rule, which for sufficiently
regular integrands exhibits second-order convergence with respect to the
maximum diameter of the subsets. As an application we show how this approach,
combined with a singularity-subtraction technique, can be used to accurately
evaluate the singular double integrals that arise in Hausdorff-measure Galerkin
boundary element methods for acoustic wave scattering by fractal screens
SPURIOUS QUASI-RESONANCES IN BOUNDARY INTEGRAL EQUATIONS FOR THE HELMHOLTZ TRANSMISSION PROBLEM
No full textWe consider the Helmholtz transmission problem with piecewise-constant material coefficients and the standard associated direct boundary integral equations. For certain coefficients and geometries, the norms of the inverses of the boundary integral operators grow rapidly through an increasing sequence of frequencies, even though this is not the case for the solution operator of the transmission problem; we call this phenomenon that of spurious quasi-resonances. We give a rigorous explanation of why and when spurious quasi-resonances occur and propose modified boundary integral equations that are not affected by the
A space-time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients
No full textTrefftz methods are high-order Galerkin schemes in which all discrete
functions are elementwise solution of the PDE to be approximated. They are
viable only when the PDE is linear and its coefficients are piecewise constant.
We introduce a 'quasi-Trefftz' discontinuous Galerkin method for the
discretisation of the acoustic wave equation with piecewise-smooth wavespeed:
the discrete functions are elementwise approximate PDE solutions. We show that
the new discretisation enjoys the same excellent approximation properties as
the classical Trefftz one, and prove stability and high-order convergence of
the DG scheme. We introduce polynomial basis functions for the new discrete
spaces and describe a simple algorithm to compute them. The technique we
propose is inspired by the generalised plane waves previously developed for
time-harmonic problems with variable coefficients; it turns out that in the
case of the time-domain wave equation under consideration the quasi-Trefftz
approach allows for polynomial basis functions.Comment: 29 pages, 11 figures, 4 table
Space-time discontinuous Galerkin approximation of acoustic waves with point singularities
No full textWe develop a convergence theory of space-time discretizations for the linear,
2nd-order wave equation in polygonal domains ,
possibly occupied by piecewise homogeneous media with different propagation
speeds. Building on an unconditionally stable space-time DG formulation
developed in [Moiola, Perugia 2018], we (a) prove optimal convergence rates for
the space-time scheme with local isotropic corner mesh refinement on the
spatial domain, and (b) demonstrate numerically optimal convergence rates of a
suitable \emph{sparse} space-time version of the DG scheme. The latter scheme
is based on the so-called \emph{combination formula}, in conjunction with a
family of anisotropic space-time DG-discretizations. It results in
optimal-order convergent schemes, also in domains with corners, with a number
of degrees of freedom that scales essentially like the DG solution of one
stationary elliptic problem in on the finest spatial grid. Numerical
experiments for both smooth and singular solutions support convergence rate
optimality on spatially refined meshes of the full and sparse space-time DG
schemes.Comment: 38 pages, 8 figure
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Analysis of the internal electric fields of pristine ice crystals and aggregate snowflakes, and their effect on scattering
No full textThe discrete dipole approximation is used to explore the internal electric fields of plane-wave-illuminated ice particles, and analyse their differential scattering cross sections. The results are displayed for monocrystals and aggregates of size parameters x=2 and x=10. We show that the field is relatively uniform for x=2, but for monocrystals of x=10 there is a complex internal structure. For a hexagonal plate, this structure is a combination of two components: a "distorted" plane wave, with wavefronts aligned perpendicular to the incident wave close to the centre of the plate, and curved forward near the particle boundary; and a standing wave, internally reflected around the perimeter. The former is due to the transverse component of the field i.e., the component perpendicular to the incident wave, and the latter is due to the component parallel to the incident direction. Focussing of the field towards the forward side of the particle is observed. As the particle complexity is increased due to aggregation, the field becomes smoother and less focussing is seen. For complex aggregates, the individual monomers act independently of one another, suggesting simplified methods of calculating scattering from such particles. The influence of the internal fields on far-field scattering is explored. It is demonstrated that scattering in the forward and backward directions is dominated by the transverse component. The parallel component contributes to sidescattering, with its influence on total scattering decreasing with particle complexity. We propose that this is due to the inability of complex particles to maintain a standing wave, diminishing much of the sidescattering observed for monocrystals. Comparisons of the far-field scattering properties of complex aggregates using the discrete dipole and Rayleigh-Gans approximations are also presented for x=2 and x=10, along with results obtained using a soft sphere approximation
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